hypothesis testing p value 0

P-Value And Statistical Significance: What It Is & Why It Matters

Saul McLeod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul McLeod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

Learn about our Editorial Process

Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

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The p-value in statistics quantifies the evidence against a null hypothesis. A low p-value suggests data is inconsistent with the null, potentially favoring an alternative hypothesis. Common significance thresholds are 0.05 or 0.01.

Hypothesis testing

When you perform a statistical test, a p-value helps you determine the significance of your results in relation to the null hypothesis.

The null hypothesis (H0) states no relationship exists between the two variables being studied (one variable does not affect the other). It states the results are due to chance and are not significant in supporting the idea being investigated. Thus, the null hypothesis assumes that whatever you try to prove did not happen.

The alternative hypothesis (Ha or H1) is the one you would believe if the null hypothesis is concluded to be untrue.

The alternative hypothesis states that the independent variable affected the dependent variable, and the results are significant in supporting the theory being investigated (i.e., the results are not due to random chance).

What a p-value tells you

A p-value, or probability value, is a number describing how likely it is that your data would have occurred by random chance (i.e., that the null hypothesis is true).

The level of statistical significance is often expressed as a p-value between 0 and 1.

The smaller the p -value, the less likely the results occurred by random chance, and the stronger the evidence that you should reject the null hypothesis.

Remember, a p-value doesn’t tell you if the null hypothesis is true or false. It just tells you how likely you’d see the data you observed (or more extreme data) if the null hypothesis was true. It’s a piece of evidence, not a definitive proof.

Example: Test Statistic and p-Value

Suppose you’re conducting a study to determine whether a new drug has an effect on pain relief compared to a placebo. If the new drug has no impact, your test statistic will be close to the one predicted by the null hypothesis (no difference between the drug and placebo groups), and the resulting p-value will be close to 1. It may not be precisely 1 because real-world variations may exist. Conversely, if the new drug indeed reduces pain significantly, your test statistic will diverge further from what’s expected under the null hypothesis, and the p-value will decrease. The p-value will never reach zero because there’s always a slim possibility, though highly improbable, that the observed results occurred by random chance.

P-value interpretation

The significance level (alpha) is a set probability threshold (often 0.05), while the p-value is the probability you calculate based on your study or analysis.

A p-value less than or equal to your significance level (typically ≤ 0.05) is statistically significant.

A p-value less than or equal to a predetermined significance level (often 0.05 or 0.01) indicates a statistically significant result, meaning the observed data provide strong evidence against the null hypothesis.

This suggests the effect under study likely represents a real relationship rather than just random chance.

For instance, if you set α = 0.05, you would reject the null hypothesis if your p -value ≤ 0.05. 

It indicates strong evidence against the null hypothesis, as there is less than a 5% probability the null is correct (and the results are random).

Therefore, we reject the null hypothesis and accept the alternative hypothesis.

Example: Statistical Significance

Upon analyzing the pain relief effects of the new drug compared to the placebo, the computed p-value is less than 0.01, which falls well below the predetermined alpha value of 0.05. Consequently, you conclude that there is a statistically significant difference in pain relief between the new drug and the placebo.

What does a p-value of 0.001 mean?

A p-value of 0.001 is highly statistically significant beyond the commonly used 0.05 threshold. It indicates strong evidence of a real effect or difference, rather than just random variation.

Specifically, a p-value of 0.001 means there is only a 0.1% chance of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is correct.

Such a small p-value provides strong evidence against the null hypothesis, leading to rejecting the null in favor of the alternative hypothesis.

A p-value more than the significance level (typically p > 0.05) is not statistically significant and indicates strong evidence for the null hypothesis.

This means we retain the null hypothesis and reject the alternative hypothesis. You should note that you cannot accept the null hypothesis; we can only reject it or fail to reject it.

Note : when the p-value is above your threshold of significance,  it does not mean that there is a 95% probability that the alternative hypothesis is true.

One-Tailed Test

Two-Tailed Test

How do you calculate the p-value ?

Most statistical software packages like R, SPSS, and others automatically calculate your p-value. This is the easiest and most common way.

Online resources and tables are available to estimate the p-value based on your test statistic and degrees of freedom.

These tables help you understand how often you would expect to see your test statistic under the null hypothesis.

Understanding the Statistical Test:

Different statistical tests are designed to answer specific research questions or hypotheses. Each test has its own underlying assumptions and characteristics.

For example, you might use a t-test to compare means, a chi-squared test for categorical data, or a correlation test to measure the strength of a relationship between variables.

Be aware that the number of independent variables you include in your analysis can influence the magnitude of the test statistic needed to produce the same p-value.

This factor is particularly important to consider when comparing results across different analyses.

Example: Choosing a Statistical Test

If you’re comparing the effectiveness of just two different drugs in pain relief, a two-sample t-test is a suitable choice for comparing these two groups. However, when you’re examining the impact of three or more drugs, it’s more appropriate to employ an Analysis of Variance ( ANOVA) . Utilizing multiple pairwise comparisons in such cases can lead to artificially low p-values and an overestimation of the significance of differences between the drug groups.

How to report

A statistically significant result cannot prove that a research hypothesis is correct (which implies 100% certainty).

Instead, we may state our results “provide support for” or “give evidence for” our research hypothesis (as there is still a slight probability that the results occurred by chance and the null hypothesis was correct – e.g., less than 5%).

Example: Reporting the results

In our comparison of the pain relief effects of the new drug and the placebo, we observed that participants in the drug group experienced a significant reduction in pain ( M = 3.5; SD = 0.8) compared to those in the placebo group ( M = 5.2; SD  = 0.7), resulting in an average difference of 1.7 points on the pain scale (t(98) = -9.36; p < 0.001).

The 6th edition of the APA style manual (American Psychological Association, 2010) states the following on the topic of reporting p-values:

“When reporting p values, report exact p values (e.g., p = .031) to two or three decimal places. However, report p values less than .001 as p < .001.

The tradition of reporting p values in the form p < .10, p < .05, p < .01, and so forth, was appropriate in a time when only limited tables of critical values were available.” (p. 114)

• Do not use 0 before the decimal point for the statistical value p as it cannot equal 1. In other words, write p = .001 instead of p = 0.001.
• Please pay attention to issues of italics ( p is always italicized) and spacing (either side of the = sign).
• p = .000 (as outputted by some statistical packages such as SPSS) is impossible and should be written as p < .001.
• The opposite of significant is “nonsignificant,” not “insignificant.”

Why is the p -value not enough?

A lower p-value  is sometimes interpreted as meaning there is a stronger relationship between two variables.

However, statistical significance means that it is unlikely that the null hypothesis is true (less than 5%).

To understand the strength of the difference between the two groups (control vs. experimental) a researcher needs to calculate the effect size .

When do you reject the null hypothesis?

In statistical hypothesis testing, you reject the null hypothesis when the p-value is less than or equal to the significance level (α) you set before conducting your test. The significance level is the probability of rejecting the null hypothesis when it is true. Commonly used significance levels are 0.01, 0.05, and 0.10.

Remember, rejecting the null hypothesis doesn’t prove the alternative hypothesis; it just suggests that the alternative hypothesis may be plausible given the observed data.

The p -value is conditional upon the null hypothesis being true but is unrelated to the truth or falsity of the alternative hypothesis.

What does p-value of 0.05 mean?

If your p-value is less than or equal to 0.05 (the significance level), you would conclude that your result is statistically significant. This means the evidence is strong enough to reject the null hypothesis in favor of the alternative hypothesis.

Are all p-values below 0.05 considered statistically significant?

No, not all p-values below 0.05 are considered statistically significant. The threshold of 0.05 is commonly used, but it’s just a convention. Statistical significance depends on factors like the study design, sample size, and the magnitude of the observed effect.

A p-value below 0.05 means there is evidence against the null hypothesis, suggesting a real effect. However, it’s essential to consider the context and other factors when interpreting results.

Researchers also look at effect size and confidence intervals to determine the practical significance and reliability of findings.

How does sample size affect the interpretation of p-values?

Sample size can impact the interpretation of p-values. A larger sample size provides more reliable and precise estimates of the population, leading to narrower confidence intervals.

With a larger sample, even small differences between groups or effects can become statistically significant, yielding lower p-values. In contrast, smaller sample sizes may not have enough statistical power to detect smaller effects, resulting in higher p-values.

Therefore, a larger sample size increases the chances of finding statistically significant results when there is a genuine effect, making the findings more trustworthy and robust.

Can a non-significant p-value indicate that there is no effect or difference in the data?

No, a non-significant p-value does not necessarily indicate that there is no effect or difference in the data. It means that the observed data do not provide strong enough evidence to reject the null hypothesis.

There could still be a real effect or difference, but it might be smaller or more variable than the study was able to detect.

Other factors like sample size, study design, and measurement precision can influence the p-value. It’s important to consider the entire body of evidence and not rely solely on p-values when interpreting research findings.

Can P values be exactly zero?

While a p-value can be extremely small, it cannot technically be absolute zero. When a p-value is reported as p = 0.000, the actual p-value is too small for the software to display. This is often interpreted as strong evidence against the null hypothesis. For p values less than 0.001, report as p < .001

Further Information

• P Value Calculator From T Score
• P-Value Calculator For Chi-Square
• P-values and significance tests (Kahn Academy)
• Hypothesis testing and p-values (Kahn Academy)
• Wasserstein, R. L., Schirm, A. L., & Lazar, N. A. (2019). Moving to a world beyond “ p “< 0.05”.
• Criticism of using the “ p “< 0.05”.
• Publication manual of the American Psychological Association
• Statistics for Psychology Book Download

Bland, J. M., & Altman, D. G. (1994). One and two sided tests of significance: Authors’ reply.  BMJ: British Medical Journal ,  309 (6958), 874.

Goodman, S. N., & Royall, R. (1988). Evidence and scientific research.  American Journal of Public Health ,  78 (12), 1568-1574.

Goodman, S. (2008, July). A dirty dozen: twelve p-value misconceptions . In  Seminars in hematology  (Vol. 45, No. 3, pp. 135-140). WB Saunders.

Lang, J. M., Rothman, K. J., & Cann, C. I. (1998). That confounded P-value.  Epidemiology (Cambridge, Mass.) ,  9 (1), 7-8.

P-Value in Statistical Hypothesis Tests: What is it?

P value definition.

A p value is used in hypothesis testing to help you support or reject the null hypothesis . The p value is the evidence against a null hypothesis . The smaller the p-value, the stronger the evidence that you should reject the null hypothesis.

P values are expressed as decimals although it may be easier to understand what they are if you convert them to a percentage . For example, a p value of 0.0254 is 2.54%. This means there is a 2.54% chance your results could be random (i.e. happened by chance). That’s pretty tiny. On the other hand, a large p-value of .9(90%) means your results have a 90% probability of being completely random and not due to anything in your experiment. Therefore, the smaller the p-value, the more important (“ significant “) your results.

When you run a hypothesis test , you compare the p value from your test to the alpha level you selected when you ran the test. Alpha levels can also be written as percentages.

P Value vs Alpha level

Alpha levels are controlled by the researcher and are related to confidence levels . You get an alpha level by subtracting your confidence level from 100%. For example, if you want to be 98 percent confident in your research, the alpha level would be 2% (100% – 98%). When you run the hypothesis test, the test will give you a value for p. Compare that value to your chosen alpha level. For example, let’s say you chose an alpha level of 5% (0.05). If the results from the test give you:

• A small p (≤ 0.05), reject the null hypothesis . This is strong evidence that the null hypothesis is invalid.
• A large p (> 0.05) means the alternate hypothesis is weak, so you do not reject the null.

P Values and Critical Values

What if I Don’t Have an Alpha Level?

In an ideal world, you’ll have an alpha level. But if you do not, you can still use the following rough guidelines in deciding whether to support or reject the null hypothesis:

• If p > .10 → “not significant”
• If p ≤ .10 → “marginally significant”
• If p ≤ .05 → “significant”
• If p ≤ .01 → “highly significant.”

How to Calculate a P Value on the TI 83

Example question: The average wait time to see an E.R. doctor is said to be 150 minutes. You think the wait time is actually less. You take a random sample of 30 people and find their average wait is 148 minutes with a standard deviation of 5 minutes. Assume the distribution is normal. Find the p value for this test.

• Press STAT then arrow over to TESTS.
• Press ENTER for Z-Test .
• Arrow over to Stats. Press ENTER.
• Arrow down to μ0 and type 150. This is our null hypothesis mean.
• Arrow down to σ. Type in your std dev: 5.
• Arrow down to xbar. Type in your sample mean : 148.
• Arrow down to n. Type in your sample size : 30.
• Arrow to <μ0 for a left tail test . Press ENTER.
• Arrow down to Calculate. Press ENTER. P is given as .014, or about 1%.

The probability that you would get a sample mean of 148 minutes is tiny, so you should reject the null hypothesis.

Note : If you don’t want to run a test, you could also use the TI 83 NormCDF function to get the area (which is the same thing as the probability value).

Dodge, Y. (2008). The Concise Encyclopedia of Statistics . Springer. Gonick, L. (1993). The Cartoon Guide to Statistics . HarperPerennial.

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Statistics By Jim

Making statistics intuitive

How Hypothesis Tests Work: Significance Levels (Alpha) and P values

By Jim Frost 45 Comments

Hypothesis testing is a vital process in inferential statistics where the goal is to use sample data to draw conclusions about an entire population . In the testing process, you use significance levels and p-values to determine whether the test results are statistically significant.

You hear about results being statistically significant all of the time. But, what do significance levels, P values, and statistical significance actually represent? Why do we even need to use hypothesis tests in statistics?

In this post, I answer all of these questions. I use graphs and concepts to explain how hypothesis tests function in order to provide a more intuitive explanation. This helps you move on to understanding your statistical results.

Hypothesis Test Example Scenario

To start, I’ll demonstrate why we need to use hypothesis tests using an example.

A researcher is studying fuel expenditures for families and wants to determine if the monthly cost has changed since last year when the average was $260 per month. The researcher draws a random sample of 25 families and enters their monthly costs for this year into statistical software. You can download the CSV data file: FuelsCosts . Below are the descriptive statistics for this year.

We’ll build on this example to answer the research question and show how hypothesis tests work.

Descriptive Statistics Alone Won’t Answer the Question

The researcher collected a random sample and found that this year’s sample mean (330.6) is greater than last year’s mean (260). Why perform a hypothesis test at all? We can see that this year’s mean is higher by $70! Isn’t that different?

Regrettably, the situation isn’t as clear as you might think because we’re analyzing a sample instead of the full population. There are huge benefits when working with samples because it is usually impossible to collect data from an entire population. However, the tradeoff for working with a manageable sample is that we need to account for sample error.

The sampling error is the gap between the sample statistic and the population parameter. For our example, the sample statistic is the sample mean, which is 330.6. The population parameter is μ, or mu, which is the average of the entire population. Unfortunately, the value of the population parameter is not only unknown but usually unknowable. Learn more about Sampling Error .

We obtained a sample mean of 330.6. However, it’s conceivable that, due to sampling error, the mean of the population might be only 260. If the researcher drew another random sample, the next sample mean might be closer to 260. It’s impossible to assess this possibility by looking at only the sample mean. Hypothesis testing is a form of inferential statistics that allows us to draw conclusions about an entire population based on a representative sample. We need to use a hypothesis test to determine the likelihood of obtaining our sample mean if the population mean is 260.

Background information : The Difference between Descriptive and Inferential Statistics and Populations, Parameters, and Samples in Inferential Statistics

A Sampling Distribution Determines Whether Our Sample Mean is Unlikely

It is very unlikely for any sample mean to equal the population mean because of sample error. In our case, the sample mean of 330.6 is almost definitely not equal to the population mean for fuel expenditures.

If we could obtain a substantial number of random samples and calculate the sample mean for each sample, we’d observe a broad spectrum of sample means. We’d even be able to graph the distribution of sample means from this process.

This type of distribution is called a sampling distribution. You obtain a sampling distribution by drawing many random samples of the same size from the same population. Why the heck would we do this?

Because sampling distributions allow you to determine the likelihood of obtaining your sample statistic and they’re crucial for performing hypothesis tests.

Luckily, we don’t need to go to the trouble of collecting numerous random samples! We can estimate the sampling distribution using the t-distribution, our sample size, and the variability in our sample.

We want to find out if the average fuel expenditure this year (330.6) is different from last year (260). To answer this question, we’ll graph the sampling distribution based on the assumption that the mean fuel cost for the entire population has not changed and is still 260. In statistics, we call this lack of effect, or no change, the null hypothesis . We use the null hypothesis value as the basis of comparison for our observed sample value.

Sampling distributions and t-distributions are types of probability distributions.

Related posts : Sampling Distributions and Understanding Probability Distributions

Graphing our Sample Mean in the Context of the Sampling Distribution

The graph below shows which sample means are more likely and less likely if the population mean is 260. We can place our sample mean in this distribution. This larger context helps us see how unlikely our sample mean is if the null hypothesis is true (μ = 260).

The graph displays the estimated distribution of sample means. The most likely values are near 260 because the plot assumes that this is the true population mean. However, given random sampling error, it would not be surprising to observe sample means ranging from 167 to 352. If the population mean is still 260, our observed sample mean (330.6) isn’t the most likely value, but it’s not completely implausible either.

The Role of Hypothesis Tests

The sampling distribution shows us that we are relatively unlikely to obtain a sample of 330.6 if the population mean is 260. Is our sample mean so unlikely that we can reject the notion that the population mean is 260?

In statistics, we call this rejecting the null hypothesis. If we reject the null for our example, the difference between the sample mean (330.6) and 260 is statistically significant. In other words, the sample data favor the hypothesis that the population average does not equal 260.

However, look at the sampling distribution chart again. Notice that there is no special location on the curve where you can definitively draw this conclusion. There is only a consistent decrease in the likelihood of observing sample means that are farther from the null hypothesis value. Where do we decide a sample mean is far away enough?

To answer this question, we’ll need more tools—hypothesis tests! The hypothesis testing procedure quantifies the unusualness of our sample with a probability and then compares it to an evidentiary standard. This process allows you to make an objective decision about the strength of the evidence.

We’re going to add the tools we need to make this decision to the graph—significance levels and p-values!

These tools allow us to test these two hypotheses:

• Null hypothesis: The population mean equals the null hypothesis mean (260).
• Alternative hypothesis: The population mean does not equal the null hypothesis mean (260).

Related post : Hypothesis Testing Overview

What are Significance Levels (Alpha)?

A significance level, also known as alpha or α, is an evidentiary standard that a researcher sets before the study. It defines how strongly the sample evidence must contradict the null hypothesis before you can reject the null hypothesis for the entire population. The strength of the evidence is defined by the probability of rejecting a null hypothesis that is true. In other words, it is the probability that you say there is an effect when there is no effect.

For instance, a significance level of 0.05 signifies a 5% risk of deciding that an effect exists when it does not exist.

Lower significance levels require stronger sample evidence to be able to reject the null hypothesis. For example, to be statistically significant at the 0.01 significance level requires more substantial evidence than the 0.05 significance level. However, there is a tradeoff in hypothesis tests. Lower significance levels also reduce the power of a hypothesis test to detect a difference that does exist.

The technical nature of these types of questions can make your head spin. A picture can bring these ideas to life!

To learn a more conceptual approach to significance levels, see my post about Understanding Significance Levels .

Graphing Significance Levels as Critical Regions

On the probability distribution plot, the significance level defines how far the sample value must be from the null value before we can reject the null. The percentage of the area under the curve that is shaded equals the probability that the sample value will fall in those regions if the null hypothesis is correct.

To represent a significance level of 0.05, I’ll shade 5% of the distribution furthest from the null value.

The two shaded regions in the graph are equidistant from the central value of the null hypothesis. Each region has a probability of 0.025, which sums to our desired total of 0.05. These shaded areas are called the critical region for a two-tailed hypothesis test.

The critical region defines sample values that are improbable enough to warrant rejecting the null hypothesis. If the null hypothesis is correct and the population mean is 260, random samples (n=25) from this population have means that fall in the critical region 5% of the time.

Our sample mean is statistically significant at the 0.05 level because it falls in the critical region.

Related posts : One-Tailed and Two-Tailed Tests Explained , What Are Critical Values? , and T-distribution Table of Critical Values

Comparing Significance Levels

Let’s redo this hypothesis test using the other common significance level of 0.01 to see how it compares.

This time the sum of the two shaded regions equals our new significance level of 0.01. The mean of our sample does not fall within with the critical region. Consequently, we fail to reject the null hypothesis. We have the same exact sample data, the same difference between the sample mean and the null hypothesis value, but a different test result.

What happened? By specifying a lower significance level, we set a higher bar for the sample evidence. As the graph shows, lower significance levels move the critical regions further away from the null value. Consequently, lower significance levels require more extreme sample means to be statistically significant.

You must set the significance level before conducting a study. You don’t want the temptation of choosing a level after the study that yields significant results. The only reason I compared the two significance levels was to illustrate the effects and explain the differing results.

The graphical version of the 1-sample t-test we created allows us to determine statistical significance without assessing the P value. Typically, you need to compare the P value to the significance level to make this determination.

Related post : Step-by-Step Instructions for How to Do t-Tests in Excel

What Are P values?

P values are the probability that a sample will have an effect at least as extreme as the effect observed in your sample if the null hypothesis is correct.

This tortuous, technical definition for P values can make your head spin. Let’s graph it!

First, we need to calculate the effect that is present in our sample. The effect is the distance between the sample value and null value: 330.6 – 260 = 70.6. Next, I’ll shade the regions on both sides of the distribution that are at least as far away as 70.6 from the null (260 +/- 70.6). This process graphs the probability of observing a sample mean at least as extreme as our sample mean.

The total probability of the two shaded regions is 0.03112. If the null hypothesis value (260) is true and you drew many random samples, you’d expect sample means to fall in the shaded regions about 3.1% of the time. In other words, you will observe sample effects at least as large as 70.6 about 3.1% of the time if the null is true. That’s the P value!

Learn more about How to Find the P Value .

Using P values and Significance Levels Together

If your P value is less than or equal to your alpha level, reject the null hypothesis.

The P value results are consistent with our graphical representation. The P value of 0.03112 is significant at the alpha level of 0.05 but not 0.01. Again, in practice, you pick one significance level before the experiment and stick with it!

Using the significance level of 0.05, the sample effect is statistically significant. Our data support the alternative hypothesis, which states that the population mean doesn’t equal 260. We can conclude that mean fuel expenditures have increased since last year.

P values are very frequently misinterpreted as the probability of rejecting a null hypothesis that is actually true. This interpretation is wrong! To understand why, please read my post: How to Interpret P-values Correctly .

Discussion about Statistically Significant Results

Hypothesis tests determine whether your sample data provide sufficient evidence to reject the null hypothesis for the entire population. To perform this test, the procedure compares your sample statistic to the null value and determines whether it is sufficiently rare. “Sufficiently rare” is defined in a hypothesis test by:

• Assuming that the null hypothesis is true—the graphs center on the null value.
• The significance (alpha) level—how far out from the null value is the critical region?
• The sample statistic—is it within the critical region?

There is no special significance level that correctly determines which studies have real population effects 100% of the time. The traditional significance levels of 0.05 and 0.01 are attempts to manage the tradeoff between having a low probability of rejecting a true null hypothesis and having adequate power to detect an effect if one actually exists.

The significance level is the rate at which you incorrectly reject null hypotheses that are actually true ( type I error ). For example, for all studies that use a significance level of 0.05 and the null hypothesis is correct, you can expect 5% of them to have sample statistics that fall in the critical region. When this error occurs, you aren’t aware that the null hypothesis is correct, but you’ll reject it because the p-value is less than 0.05.

This error does not indicate that the researcher made a mistake. As the graphs show, you can observe extreme sample statistics due to sample error alone. It’s the luck of the draw!

Related posts : Statistical Significance: Definition & Meaning and Types of Errors in Hypothesis Testing

Hypothesis tests are crucial when you want to use sample data to make conclusions about a population because these tests account for sample error. Using significance levels and P values to determine when to reject the null hypothesis improves the probability that you will draw the correct conclusion.

Keep in mind that statistical significance doesn’t necessarily mean that the effect is important in a practical, real-world sense. For more information, read my post about Practical vs. Statistical Significance .

If you like this post, read the companion post: How Hypothesis Tests Work: Confidence Intervals and Confidence Levels .

You can also read my other posts that describe how other tests work:

• How t-Tests Work
• How the F-test works in ANOVA
• How Chi-Squared Tests of Independence Work

To see an alternative approach to traditional hypothesis testing that does not use probability distributions and test statistics, learn about bootstrapping in statistics !

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Reader Interactions

December 11, 2022 at 10:56 am

For very easy concept about level of significance & p-value 1.Teacher has given a one assignment to student & asked how many error you have doing this assignment? Student reply, he can has error ≤ 5% (it is level of significance). After completion of assignment, teacher checked his error which is ≤ 5% (may be 4% or 3% or 2% even less, it is p-value) it means his results are significant. Otherwise he has error > 5% (may be 6% or 7% or 8% even more, it is p-value) it means his results are non-significant. 2. Teacher has given a one assignment to student & asked how many error you have doing this assignment? Student reply, he can has error ≤ 1% (it is level of significance). After completion of assignment, teacher checked his error which is ≤ 1% (may be 0.9% or 0.8% or 0.7% even less, it is p-value) it means his results are significant. Otherwise he has error > 1% (may be 1.1% or 1.5% or 2% even more, it is p-value) it means his results are non-significant. p-value is significant or not mainly dependent upon the level of significance.

December 11, 2022 at 7:50 pm

I think that approach helps explain how to determine statistical significance–is the p-value less than or equal to the significance level. However, it doesn’t really explain what statistical significance means. I find that comparing the p-value to the significance level is the easy part. Knowing what it means and how to choose your significance level is the harder part!

December 3, 2022 at 5:54 pm

What would you say to someone who believes that a p-value higher than the level of significance (alpha) means the null hypothesis has been proven? Should you support that statement or deny it?

December 3, 2022 at 10:18 pm

Hi Emmanuel,

When the p-value is greater than the significance level, you fail to reject the null hypothesis . That is different than proving it. To learn why and what it means, click the link to read a post that I’ve written that will answer your question!

April 19, 2021 at 12:27 am

Thank you so much Sir

April 18, 2021 at 2:37 pm

Hi sir, your blogs are much more helpful for clearing the concepts of statistics, as a researcher I find them much more useful. I have some quarries:

1. In many research papers I have seen authors using the statement ” means or values are statically at par at p = 0.05″ when they do some pair wise comparison between the treatments (a kind of post hoc) using some value of CD (critical difference) or we can say LSD which is calculated using alpha not using p. So with this article I think this should be alpha =0.05 or 5%, not p = 0.05 earlier I thought p and alpha are same. p it self is compared with alpha 0.05. Correct me if I am wrong.

2. When we can draw a conclusion using critical value based on critical values (CV) which is based on alpha values in different tests (e.g. in F test CV is at F (0.05, t-1, error df) when alpha is 0.05 which is table value of F and is compared with F calculated for drawing the conclusion); then why we go for p values, and draw a conclusion based on p values, even many online software do not give p value, they just mention CD (LSD)

3. can you please help me in interpreting interaction in two factor analysis (Factor A X Factor b) in Anova.

Thank You so much!

(Commenting again as I have not seen my comment in comment list; don’t know why)

April 18, 2021 at 10:57 pm

Hi Himanshu,

I manually approve comments so there will be some time lag involved before they show up.

Regarding your first question, yes, you’re correct. Test results are significant at particular significance levels or alpha. They should not use p to define the significance level. You’re also correct in that you compare p to alpha.

Critical values are a different (but related) approach for determining significance. It was more common before computer analysis took off because it reduced the calculations. Using this approach in its simplest form, you only know whether a result is significant or not at the given alpha. You just determine whether the test statistic falls within a critical region to determine statistical significance or not significant. However, it is ok to supplement this type of result with the actual p-value. Knowing the precise p-value provides additional information that significant/not significant does not provide. The critical value and p-value approaches will always agree too. For more information about why the exact p-value is useful, read my post about Five Tips for Interpreting P-values .

Finally, I’ve written about two-way ANOVA in my post, How to do Two-Way ANOVA in Excel . Additionally, I write about it in my Hypothesis Testing ebook .

January 28, 2021 at 3:12 pm

Thank you for your answer, Jim, I really appreciate it. I’m taking a Coursera stats course and online learning without being able to ask questions of a real teacher is not my forte!

You’re right, I don’t think I’m ready for that calculation! However, I think I’m struggling with something far more basic, perhaps even the interpretation of the t-table? I’m just not sure how you came up with the p-value as .03112, with the 24 degrees of freedom. When I pull up a t-table and look at the 24-degrees of freedom row, I’m not sure how any of those numbers correspond with your answer? Either the single tail of 0.01556 or the combined of 0.03112. What am I not getting? (which, frankly, could be a lot!!) Again, thank you SO much for your time.

January 28, 2021 at 11:19 pm

Ah ok, I see! First, let me point you to several posts I’ve written about t-values and the t-distribution. I don’t cover those in this post because I wanted to present a simplified version that just uses the data in its regular units. The basic idea is that the hypothesis tests actually convert all your raw data down into one value for a test statistic, such as the t-value. And then it uses that test statistic to determine whether your results are statistically significant. To be significant, the t-value must exceed a critical value, which is what you lookup in the table. Although, nowadays you’d typically let your software just tell you.

So, read the following two posts, which covers several aspects of t-values and distributions. And then if you have more questions after that, you can post them. But, you’ll have a lot more information about them and probably some of your questions will be answered! T-values T-distributions

January 27, 2021 at 3:10 pm

Jim, just found your website and really appreciate your thoughtful, thorough way of explaining things. I feel very dumb, but I’m struggling with p-values and was hoping you could help me.

Here’s the section that’s getting me confused:

“First, we need to calculate the effect that is present in our sample. The effect is the distance between the sample value and null value: 330.6 – 260 = 70.6. Next, I’ll shade the regions on both sides of the distribution that are at least as far away as 70.6 from the null (260 +/- 70.6). This process graphs the probability of observing a sample mean at least as extreme as our sample mean.

** I’m good up to this point. Draw the picture, do the subtraction, shade the regions. BUT, I’m not sure how to figure out the area of the shaded region — even with a T-table. When I look at the T-table on 24 df, I’m not sure what to do with those numbers, as none of them seem to correspond in any way to what I’m looking at in the problem. In the end, I have no idea how you calculated each shaded area being 0.01556.

I feel like there’s a (very simple) step that everyone else knows how to do, but for some reason I’m missing it.

Again, dumb question, but I’d love your help clarifying that.

thank you, Sara

January 27, 2021 at 9:51 pm

That’s not a dumb question at all. I actually don’t show or explain the calculations for figuring out the area. The reason for that is the same reason why students never calculate the critical t-values for their test, instead you look them up in tables or use statistical software. The common reason for all that is because calculating these values is extremely complicated! It’s best to let software do that for you or, when looking critical values, use the tables!

The principal though is that percentage of the area under the curve equals the probability that values will fall within that range.

And then, for this example, you’d need to figure out the area under the curve for particular ranges!

January 15, 2021 at 10:57 am

HI Jim, I have a question related to Hypothesis test.. in Medical imaging, there are different way to measure signal intensity (from a tumor lesion for example). I tested for the same 100 patients 4 different ways to measure tumor captation to a injected dose. So for the 100 patients, i got 4 linear regression (relation between injected dose and measured quantity at tumor sites) = so an output of 4 equations Condition A output = -0,034308 + 0,0006602*input Condition B output = 0,0117631 + 0,0005425*input Condition C output = 0,0087871 + 0,0005563*input Condition D output = 0,001911 + 0,0006255*input

My question : i want to compare the 4 methods to find the best one (compared to others) : do Hypothesis test good to me… and if Yes, i do not find test to perform it. Can you suggest me a software. I uselly used JMP for my stats… but open to other softwares…

THank for your time G

November 16, 2020 at 5:42 am

Thank you very much for writing about this topic!

Your explanation made more sense to me about: Why we reject Null Hypothesis when p value < significance level

Kind greetings, Jalal

September 25, 2020 at 1:04 pm

Hi Jim, Your explanations are so helpful! Thank you. I wondered about your first graph. I see that the mean of the graph is 260 from the null hypothesis, and it looks like the standard deviation of the graph is about 31. Where did you get 31 from? Thank you

September 25, 2020 at 4:08 pm

Hi Michelle,

That is a great question. Very observant. And it gets to how these tests work. The hypothesis test that I’m illustrating here is the one-sample t-test. And this graph illustrates the sampling distribution for the t-test. T-tests use the t-distribution to determine the sampling distribution. For the t-distribution, you need to specify the degrees of freedom, which entirely defines the distribution (i.e., it’s the only parameter). For 1-sample t-tests, the degrees of freedom equal the number of observations minus 1. This dataset has 25 observations. Hence, the 24 DF you see in the graph.

Unlike the normal distribution, there is no standard deviation parameter. Instead, the degrees of freedom determines the spread of the curve. Typically, with t-tests, you’ll see results discussed in terms of t-values, both for your sample and for defining the critical regions. However, for this introductory example, I’ve converted the t-values into the raw data units (t-value * SE mean).

So, the standard deviation you’re seeing in the graph is a result of the spread of the underlying t-distribution that has 24 degrees of freedom and then applying the conversion from t-values to raw values.

September 10, 2020 at 8:19 am

Your blog is incredible.

I am having difficulty understanding why the phrase ‘as extreme as’ is required in the definition of p-value (“P values are the probability that a sample will have an effect at least as extreme as the effect observed in your sample if the null hypothesis is correct.”)

Why can’t P-Values simply be defined as “The probability of sample observation if the null hypothesis is correct?”

In your other blog titled ‘Interpreting P values’ you have explained p-values as “P-values indicate the believability of the devil’s advocate case that the null hypothesis is correct given the sample data”. I understand (or accept) this explanation. How does one move from this definition to one that contains the phrase ‘as extreme as’?

September 11, 2020 at 5:05 pm

Thanks so much for your kind words! I’m glad that my website has been helpful!

The key to understanding the “at least as extreme” wording lies in the probability plots for p-values. Using probability plots for continuous data, you can calculate probabilities, but only for ranges of values. I discuss this in my post about understanding probability distributions . In a nutshell, we need a range of values for these probabilities because the probabilities are derived from the area under a distribution curve. A single value just produces a line on these graphs rather than an area. Those ranges are the shaded regions in the probability plots. For p-values, the range corresponds to the “at least as extreme” wording. That’s where it comes from. We need a range to calculate a probability. We can’t use the single value of the observed effect because it doesn’t produce an area under the curve.

I hope that helps! I think this is a particularly confusing part of understanding p-values that most people don’t understand.

August 7, 2020 at 5:45 pm

Hi Jim, thanks for the post.

Could you please clarify the following excerpt from ‘Graphing Significance Levels as Critical Regions’:

“The percentage of the area under the curve that is shaded equals the probability that the sample value will fall in those regions if the null hypothesis is correct.”

I’m not sure if I understood this correctly. If the sample value fall in one of the shaded regions, doesn’t mean that the null hypothesis can be rejected, hence that is not correct?

August 7, 2020 at 10:23 pm

Think of it this way. There are two basic reasons for why a sample value could fall in a critical region:

• The null hypothesis is correct and random chance caused the sample value to be unusual.
• The null hypothesis is not correct.

You don’t know which one is true. Remember, just because you reject the null hypothesis it doesn’t mean the null is false. However, by using hypothesis tests to determine statistical significance, you control the chances of #1 occurring. The rate at which #1 occurs equals your significance level. On the hand, you don’t know the probability of the sample value falling in a critical region if the alternative hypothesis is correct (#2). It depends on the precise distribution for the alternative hypothesis and you usually don’t know that, which is why you’re testing the hypotheses in the first place!

I hope I answered the question you were asking. If not, feel free to ask follow up questions. Also, this ties into how to interpret p-values . It’s not exactly straightforward. Click the link to learn more.

June 4, 2020 at 6:17 am

Hi Jim, thank you very much for your answer. You helped me a lot!

June 3, 2020 at 5:23 pm

Hi, Thanks for this post. I’ve been learning a lot with you. My question is regarding to lack of fit. The p-value of my lack of fit is really low, making my lack of fit significant, meaning my model does not fit well. Is my case a “false negative”? given that my pure error is really low, making the computation of the lack of fit low. So it means my model is good. Below I show some information, that I hope helps to clarify my question.

SumSq DF MeanSq F pValue ________ __ ________ ______ __________

Total 1246.5 18 69.25 Model 1241.7 6 206.94 514.43 9.3841e-14 . Linear 1196.6 3 398.87 991.53 1.2318e-14 . Nonlinear 45.046 3 15.015 37.326 2.3092e-06 Residual 4.8274 12 0.40228 . Lack of fit 4.7388 7 0.67698 38.238 0.0004787 . Pure error 0.088521 5 0.017704

June 3, 2020 at 7:53 pm

As you say, a low p-value for a lack of fit test indicates that the model doesn’t fit your data adequately. This is a positive result for the test, which means it can’t be a “false negative.” At best, it could be a false positive, meaning that your data actually fit model well despite the low p-value.

I’d recommend graphing the residuals and looking for patterns . There is probably a relationship between variables that you’re not modeling correctly, such as curvature or interaction effects. There’s no way to diagnose the specific nature of the lack-of-fit problem by using the statistical output. You’ll need the graphs.

If there are no patterns in the residual plots, then your lack-of-fit results might be a false positive.

I hope this helps!

May 30, 2020 at 6:23 am

First of all, I have to say there are not many resources that explain a complicated topic in an easier manner.

My question is, how do we arrive at “if p value is less than alpha, we reject the null hypothesis.”

Is this covered in a separate article I could read?

Thanks Shekhar

May 25, 2020 at 12:21 pm

Hi Jim, terrific website, blog, and after this I’m ordering your book. One of my biggest challenges is nomenclature, definitions, context, and formulating the hypotheses. Here’s one I want to double-be-sure I understand: From above you write: ” These tools allow us to test these two hypotheses:

Null hypothesis: The population mean equals the null hypothesis mean (260). Alternative hypothesis: The population mean does not equal the null hypothesis mean (260). ” I keep thinking that 260 is the population mean mu, the underlying population (that we never really know exactly) and that the Null Hypothesis is comparing mu to x-bar (the sample mean of the 25 families randomly sampled w mean = sample mean = x-bar = 330.6).

So is the following incorrect, and if so, why? Null hypothesis: The population mean mu=260 equals the null hypothesis mean x-bar (330.6). Alternative hypothesis: The population mean mu=269 does not equal the null hypothesis mean x-bar (330.6).

And my thinking is that usually the formulation of null and alternative hypotheses is “test value” = “mu current of underlying population”, whereas I read the formulation on the webpage above to be the reverse.

Any comments appreciated. Many Thanks,

May 26, 2020 at 8:56 pm

The null hypothesis states that population value equals the null value. Now, I know that’s not particularly helpful! But, the null value varies based on test and context. So, in this example, we’re setting the null value aa $260, which was the mean from the previous year. So, our null hypothesis states:

Null: the population mean (mu) = 260. Alternative: the population mean ≠ 260.

These hypothesis statements are about the population parameter. For this type of one-sample analysis, the target or reference value you specify is the null hypothesis value. Additionally, you don’t include the sample estimate in these statements, which is the X-bar portion you tacked on at the end. It’s strictly about the value of the population parameter you’re testing. You don’t know the value of the underlying distribution. However, given the mutually exclusive nature of the null and alternative hypothesis, you know one or the other is correct. The null states that mu equals 260 while the alternative states that it doesn’t equal 260. The data help you decide, which brings us to . . .

However, the procedure does compare our sample data to the null hypothesis value, which is how it determines how strong our evidence is against the null hypothesis.

I hope I answered your question. If not, please let me know!

May 8, 2020 at 6:00 pm

Really using the interpretation “In other words, you will observe sample effects at least as large as 70.6 about 3.1% of the time if the null is true”, our head seems to tie a knot. However, doing the reverse interpretation, it is much more intuitive and easier. That is, we will observe the sample effect of at least 70.6 in about 96.9% of the time, if the null is false (that is, our hypothesis is true).

May 8, 2020 at 7:25 pm

Your phrasing really isn’t any simpler. And it has the additional misfortune of being incorrect.

What you’re essentially doing is creating a one-sided confidence interval by using the p-value from a two-sided test. That’s incorrect in two ways.

• Don’t mix and match one-sided and two-sided test results.
• Confidence levels are determine by the significance level, not p-values.

So, what you need is a two-sided 95% CI (1-alpha). You could then state the results are statistically significant and you have 95% confidence that the population effect is between X and Y. If you want a lower bound as you propose, then you’ll need to use a one-sided hypothesis test with a 95% Lower Bound. That’ll give you a different value for the lower bound than the one you use.

I like confidence intervals. As I write elsewhere, I think they’re easier to understand and provide more information than a binary test result. But, you need to use them correctly!

One other point. When you are talking about p-values, it’s always under the assumption that the null hypothesis is correct. You *never* state anything about the p-value in relation to the null being false (i.e. alternative is true). But, if you want to use the type of phrasing you suggest, use it in the context of CIs and incorporate the points I cover above.

February 10, 2020 at 11:13 am

Muchas gracias profesor por compartir sus conocimientos. Un saliud especial desde Colombia.

August 6, 2019 at 11:46 pm

i found this really helpful . also can you help me out ?

I’m a little confused Can you tell me if level of significance and pvalue are comparable or not and if they are what does it mean if pvalue < LS . Do we reject the null hypothesis or do we accept the null hypothesis ?

August 7, 2019 at 12:49 am

Hi Divyanshu,

Yes, you compare the p-value to the significance level. When the p-value is less than the significance level (alpha), your results are statistically significant and you reject the null hypothesis.

I’d suggest re-reading the “Using P values and Significance Levels Together” section near the end of this post more closely. That describes the process. The next section describes what it all means.

July 1, 2019 at 4:19 am

sure.. I will use only in my class rooms that too offline with due credits to your orginal page. I will encourage my students to visit your blog . I have purchased your eBook on Regressions….immensely useful.

July 1, 2019 at 9:52 am

Hi Narasimha, that sounds perfect. Thanks for buying my ebook as well. I’m thrilled to hear that you’ve found it to be helpful!

June 28, 2019 at 6:22 am

I have benefited a lot by your writings….Can I share the same with my students in the classroom?

June 30, 2019 at 8:44 pm

Hi Narasimha,

Yes, you can certainly share with your students. Please attribute my original page. And please don’t copy whole sections of my posts onto another webpage as that can be bad with Google! Thanks!

February 11, 2019 at 7:46 pm

Hello, great site and my apologies if the answer to the following question exists already.

I’ve always wondered why we put the sampling distribution about the null hypothesis rather than simply leave it about the observed mean. I can see mathematically we are measuring the same distance from the null and basically can draw the same conclusions.

For example we take a sample (say 50 people) we gather an observation (mean wage) estimate the standard error in that observation and so can build a sampling distribution about the observed mean. That sampling distribution contains a confidence interval, where say, i am 95% confident the true mean lies (i.e. in repeated sampling the true mean would reside within this interval 95% of the time).

When i use this for a hyp-test, am i right in saying that we place the sampling dist over the reference level simply because it’s mathematically equivalent and it just seems easier to gauge how far the observation is from 0 via t-stats or its likelihood via p-values?

It seems more natural to me to look at it the other way around. leave the sampling distribution on the observed value, and then look where the null sits…if it’s too far left or right then it is unlikely the true population parameter is what we believed it to be, because if the null were true it would only occur ~ 5% of the time in repeated samples…so perhaps we need to change our opinion.

Can i interpret a hyp-test that way? Or do i have a misconception?

February 12, 2019 at 8:25 pm

The short answer is that, yes, you can draw the interval around the sample mean instead. And, that is, in fact, how you construct confidence intervals. The distance around the null hypothesis for hypothesis tests and the distance around the sample for confidence intervals are the same distance, which is why the results will always agree as long as you use corresponding alpha levels and confidence levels (e.g., alpha 0.05 with a 95% confidence level). I write about how this works in a post about confidence intervals .

I prefer confidence intervals for a number of reasons. They’ll indicate whether you have significant results if they exclude the null value and they indicate the precision of the effect size estimate. Corresponding with what you’re saying, it’s easier to gauge how far a confidence interval is from the null value (often zero) whereas a p-value doesn’t provide that information. See Practical versus Statistical Significance .

So, you don’t have any misconception at all! Just refer to it as a confidence interval rather than a hypothesis test, but, of course, they are very closely related.

January 9, 2019 at 10:37 pm

Hi Jim, Nice Article.. I have a question… I read the Central limit theorem article before this article…

Coming to this article, During almost every hypothesis test, we draw a normal distribution curve assuming there is a sampling distribution (and then we go for test statistic, p value etc…). Do we draw a normal distribution curve for hypo tests because of the central limit theorem…

Thanks in advance, Surya

January 10, 2019 at 1:57 am

These distributions are actually the t-distribution which are different from the normal distribution. T-distributions only have one parameter–the degrees of freedom. As the DF of increases, the t-distribution tightens up. Around 25 degrees of freedom, the t-distribution approximates the normal distribution. Depending on the type of t-test, this corresponds to a sample size of 26 or 27. Similarly, the sampling distribution of the means also approximate the normal distribution at around these sample sizes. With a large enough sample size, both the t-distribution and the sample distribution converge to a normal distribution regardless (largely) of the underlying population distribution. So, yes, the central limit theorem plays a strong role in this.

It’s more accurate to say that central limit theorem causes the sampling distribution of the means to converge on the same distribution that the t-test uses, which allows you to assume that the test produces valid results. But, technically, the t-test is based on the t-distribution.

Problems can occur if the underlying distribution is non-normal and you have a small sample size. In that case, the sampling distribution of the means won’t approximate the t-distribution that the t-test uses. However, the test results will assume that it does and produce results based on that–which is why it causes problems!

November 19, 2018 at 9:15 am

Dear Jim! Thank you very much for your explanation. I need your help to understand my data. I have two samples (about 300 observations) with biased distributions. I did the ttest and obtained the p-value, which is quite small. Can I draw the conclusion that the effect size is small even when the distribution of my data is not normal? Thank you

November 19, 2018 at 9:34 am

Hi Tetyana,

First, when you say that your p-value is small and that you want to “draw the conclusion that the effect size is small,” I assume that you mean statistically significant. When the p-value is low, the null hypothesis must go! In other words, you reject the null and conclude that there is a statistically significant effect–not a small effect.

Now, back to the question at hand! Yes, When you have a sufficiently large sample-size, t-tests are robust to departures from normality. For a 2-sample t-test, you should have at least 15 samples per group, which you exceed by quite a bit. So, yes, you can reliably conclude that your results are statistically significant!

You can thank the central limit theorem! 🙂

September 10, 2018 at 12:18 am

Hello Jim, I am very sorry; I have very elementary of knowledge of stats. So, would you please explain how you got a p- value of 0.03112 in the above calculation/t-test? By looking at a chart? Would you also explain how you got the information that “you will observe sample effects at least as large as 70.6 about 3.1% of the time if the null is true”?

July 6, 2018 at 7:02 am

A quick question regarding your use of two-tailed critical regions in the article above: why? I mean, what is a real-world scenario that would warrant a two-tailed test of any kind (z, t, etc.)? And if there are none, why keep using the two-tailed scenario as an example, instead of the one-tailed which is both more intuitive and applicable to most if not all practical situations. Just curious, as one person attempting to educate people on stats to another (my take on the one vs. two-tailed tests can be seen here: http://blog.analytics-toolkit.com/2017/one-tailed-two-tailed-tests-significance-ab-testing/ )

Thanks, Georgi

July 6, 2018 at 12:05 pm

There’s the appropriate time and place for both one-tailed and two-tailed tests. I plan to write a post on this issue specifically, so I’ll keep my comments here brief.

So much of statistics is context sensitive. People often want concrete rules for how to do things in statistics but that’s often hard to provide because the answer depends on the context, goals, etc. The question of whether to use a one-tailed or two-tailed test falls firmly in this category of it depends.

I did read the article you wrote. I’ll say that I can see how in the context of A/B testing specifically there might be a propensity to use one-tailed tests. You only care about improvements. There’s probably not too much downside in only caring about one direction. In fact, in a post where I compare different tests and different options , I suggest using a one-tailed test for a similar type of casing involving defects. So, I’m onboard with the idea of using one-tailed tests when they’re appropriate. However, I do think that two-tailed tests should be considered the default choice and that you need good reasons to move to a one-tailed test. Again, your A/B testing area might supply those reasons on a regular basis, but I can’t make that a blanket statement for all research areas.

I think your article mischaracterizes some of the pros and cons of both types of tests. Just a couple of for instances. In a two-tailed test, you don’t have to take the same action regardless of which direction the results are significant (example below). And, yes, you can determine the direction of the effect in a two-tailed test. You simply look at the estimated effect. Is it positive or negative?

On the other hand, I do agree that one-tailed tests don’t increase the overall Type I error. However, there is a big caveat for that. In a two-tailed test, the Type I error rate is evenly split in both tails. For a one-tailed test, the overall Type I error rate does not change, but the Type I errors are redistributed so they all occur in the direction that you are interested in rather than being split between the positive and negative directions. In other words, you’ll have twice as many Type I errors in the specific direction that you’re interested in. That’s not good.

My big concerns with one-tailed tests are that it makes it easier to obtain the results that you want to obtain. And, all of the Type I errors (false positives) are in that direction too. It’s just not a good combination.

To answer your question about when you might want to use two-tailed tests, there are plenty of reasons. For one, you might want to avoid the situation I describe above. Additionally, in a lot of scientific research, the researchers truly are interested in detecting effects in either direction for the sake of science. Even in cases with a practical application, you might want to learn about effects in either direction.

For example, I was involved in a research study that looked at the effects of an exercise intervention on bone density. The idea was that it might be a good way to prevent osteoporosis. I used a two-tailed test. Obviously, we’re hoping that there was positive effect. However, we’d be very interested in knowing whether there was a negative effect too. And, this illustrates how you can have different actions based on both directions. If there was a positive effect, you can recommend that as a good approach and try to promote its use. If there’s a negative effect, you’d issue a warning to not do that intervention. You have the potential for learning both what is good and what is bad. The extra false-positives would’ve cause problems because we’d think that there’d be health benefits for participants when those benefits don’t actually exist. Also, if we had performed only a one-tailed test and didn’t obtain significant results, we’d learn that it wasn’t a positive effect, but we would not know whether it was actually detrimental or not.

Here’s when I’d say it’s OK to use a one-tailed test. Consider a one-tailed test when you’re in situation where you truly only need to know whether an effect exists in one direction, and the extra Type I errors in that direction are an acceptable risk (false positives don’t cause problems), and there’s no benefit in determining whether an effect exists in the other direction. Those conditions really restrict when one-tailed tests are the best choice. Again, those restrictions might not be relevant for your specific field, but as for the usage of statistics as a whole, they’re absolutely crucial to consider.

On the other hand, according to this article, two-tailed tests might be important in A/B testing !

March 30, 2018 at 5:29 am

Dear Sir, please confirm if there is an inadvertent mistake in interpretation as, “We can conclude that mean fuel expenditures have increased since last year.” Our null hypothesis is =260. If found significant, it implies two possibilities – both increase and decrease. Please let us know if we are mistaken here. Many Thanks!

March 30, 2018 at 9:59 am

Hi Khalid, the null hypothesis as it is defined for this test represents the mean monthly expenditure for the previous year (260). The mean expenditure for the current year is 330.6 whereas it was 260 for the previous year. Consequently, the mean has increased from 260 to 330.7 over the course of a year. The p-value indicates that this increase is statistically significant. This finding does not suggest both an increase and a decrease–just an increase. Keep in mind that a significant result prompts us to reject the null hypothesis. So, we reject the null that the mean equals 260.

Let’s explore the other possible findings to be sure that this makes sense. Suppose the sample mean had been closer to 260 and the p-value was greater than the significance level, those results would indicate that the results were not statistically significant. The conclusion that we’d draw is that we have insufficient evidence to conclude that mean fuel expenditures have changed since the previous year.

If the sample mean was less than the null hypothesis (260) and if the p-value is statistically significant, we’d concluded that mean fuel expenditures have decreased and that this decrease is statistically significant.

When you interpret the results, you have to be sure to understand what the null hypothesis represents. In this case, it represents the mean monthly expenditure for the previous year and we’re comparing this year’s mean to it–hence our sample suggests an increase.

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• Knowledge Base

Hypothesis Testing | A Step-by-Step Guide with Easy Examples

Published on November 8, 2019 by Rebecca Bevans . Revised on June 22, 2023.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics . It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories.

There are 5 main steps in hypothesis testing:

• State your research hypothesis as a null hypothesis and alternate hypothesis (H o ) and (H a  or H 1 ).
• Collect data in a way designed to test the hypothesis.
• Perform an appropriate statistical test .
• Decide whether to reject or fail to reject your null hypothesis.
• Present the findings in your results and discussion section.

Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps.

Table of contents

Step 1: state your null and alternate hypothesis, step 2: collect data, step 3: perform a statistical test, step 4: decide whether to reject or fail to reject your null hypothesis, step 5: present your findings, other interesting articles, frequently asked questions about hypothesis testing.

After developing your initial research hypothesis (the prediction that you want to investigate), it is important to restate it as a null (H o ) and alternate (H a ) hypothesis so that you can test it mathematically.

The alternate hypothesis is usually your initial hypothesis that predicts a relationship between variables. The null hypothesis is a prediction of no relationship between the variables you are interested in.

• H 0 : Men are, on average, not taller than women. H a : Men are, on average, taller than women.

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For a statistical test to be valid , it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in.

There are a variety of statistical tests available, but they are all based on the comparison of within-group variance (how spread out the data is within a category) versus between-group variance (how different the categories are from one another).

If the between-group variance is large enough that there is little or no overlap between groups, then your statistical test will reflect that by showing a low p -value . This means it is unlikely that the differences between these groups came about by chance.

Alternatively, if there is high within-group variance and low between-group variance, then your statistical test will reflect that with a high p -value. This means it is likely that any difference you measure between groups is due to chance.

Your choice of statistical test will be based on the type of variables and the level of measurement of your collected data .

• an estimate of the difference in average height between the two groups.
• a p -value showing how likely you are to see this difference if the null hypothesis of no difference is true.

Based on the outcome of your statistical test, you will have to decide whether to reject or fail to reject your null hypothesis.

In most cases you will use the p -value generated by your statistical test to guide your decision. And in most cases, your predetermined level of significance for rejecting the null hypothesis will be 0.05 – that is, when there is a less than 5% chance that you would see these results if the null hypothesis were true.

In some cases, researchers choose a more conservative level of significance, such as 0.01 (1%). This minimizes the risk of incorrectly rejecting the null hypothesis ( Type I error ).

The results of hypothesis testing will be presented in the results and discussion sections of your research paper , dissertation or thesis .

In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p -value). In the discussion , you can discuss whether your initial hypothesis was supported by your results or not.

In the formal language of hypothesis testing, we talk about rejecting or failing to reject the null hypothesis. You will probably be asked to do this in your statistics assignments.

However, when presenting research results in academic papers we rarely talk this way. Instead, we go back to our alternate hypothesis (in this case, the hypothesis that men are on average taller than women) and state whether the result of our test did or did not support the alternate hypothesis.

If your null hypothesis was rejected, this result is interpreted as “supported the alternate hypothesis.”

These are superficial differences; you can see that they mean the same thing.

You might notice that we don’t say that we reject or fail to reject the alternate hypothesis . This is because hypothesis testing is not designed to prove or disprove anything. It is only designed to test whether a pattern we measure could have arisen spuriously, or by chance.

If we reject the null hypothesis based on our research (i.e., we find that it is unlikely that the pattern arose by chance), then we can say our test lends support to our hypothesis . But if the pattern does not pass our decision rule, meaning that it could have arisen by chance, then we say the test is inconsistent with our hypothesis .

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Normal distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Cite this Scribbr article

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9.3 - the p-value approach, example 9-4 section  .

Up until now, we have used the critical region approach in conducting our hypothesis tests. Now, let's take a look at an example in which we use what is called the P -value approach .

Among patients with lung cancer, usually, 90% or more die within three years. As a result of new forms of treatment, it is felt that this rate has been reduced. In a recent study of n = 150 lung cancer patients, y = 128 died within three years. Is there sufficient evidence at the \(\alpha = 0.05\) level, say, to conclude that the death rate due to lung cancer has been reduced?

The sample proportion is:

\(\hat{p}=\dfrac{128}{150}=0.853\)

The null and alternative hypotheses are:

\(H_0 \colon p = 0.90\) and \(H_A \colon p < 0.90\)

The test statistic is, therefore:

\(Z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}=\dfrac{0.853-0.90}{\sqrt{\dfrac{0.90(0.10)}{150}}}=-1.92\)

And, the rejection region is:

Since the test statistic Z = −1.92 < −1.645, we reject the null hypothesis. There is sufficient evidence at the \(\alpha = 0.05\) level to conclude that the rate has been reduced.

Example 9-4 (continued) Section  

What if we set the significance level \(\alpha\) = P (Type I Error) to 0.01? Is there still sufficient evidence to conclude that the death rate due to lung cancer has been reduced?

In this case, with \(\alpha = 0.01\), the rejection region is Z ≤ −2.33. That is, we reject if the test statistic falls in the rejection region defined by Z ≤ −2.33:

Because the test statistic Z = −1.92 > −2.33, we do not reject the null hypothesis. There is insufficient evidence at the \(\alpha = 0.01\) level to conclude that the rate has been reduced.

In the first part of this example, we rejected the null hypothesis when \(\alpha = 0.05\). And, in the second part of this example, we failed to reject the null hypothesis when \(\alpha = 0.01\). There must be some level of \(\alpha\), then, in which we cross the threshold from rejecting to not rejecting the null hypothesis. What is the smallest \(\alpha \text{ -level}\) that would still cause us to reject the null hypothesis?

We would, of course, reject any time the critical value was smaller than our test statistic −1.92:

That is, we would reject if the critical value were −1.645, −1.83, and −1.92. But, we wouldn't reject if the critical value were −1.93. The \(\alpha \text{ -level}\) associated with the test statistic −1.92 is called the P -value . It is the smallest \(\alpha \text{ -level}\) that would lead to rejection. In this case, the P -value is:

P ( Z < −1.92) = 0.0274

So far, all of the examples we've considered have involved a one-tailed hypothesis test in which the alternative hypothesis involved either a less than (<) or a greater than (>) sign. What happens if we weren't sure of the direction in which the proportion could deviate from the hypothesized null value? That is, what if the alternative hypothesis involved a not-equal sign (≠)? Let's take a look at an example.

What if we wanted to perform a " two-tailed " test? That is, what if we wanted to test:

\(H_0 \colon p = 0.90\) versus \(H_A \colon p \ne 0.90\)

at the \(\alpha = 0.05\) level?

Let's first consider the critical value approach . If we allow for the possibility that the sample proportion could either prove to be too large or too small, then we need to specify a threshold value, that is, a critical value, in each tail of the distribution. In this case, we divide the " significance level " \(\alpha\) by 2 to get \(\alpha/2\):

That is, our rejection rule is that we should reject the null hypothesis \(H_0 \text{ if } Z ≥ 1.96\) or we should reject the null hypothesis \(H_0 \text{ if } Z ≤ −1.96\). Alternatively, we can write that we should reject the null hypothesis \(H_0 \text{ if } |Z| ≥ 1.96\). Because our test statistic is −1.92, we just barely fail to reject the null hypothesis, because 1.92 < 1.96. In this case, we would say that there is insufficient evidence at the \(\alpha = 0.05\) level to conclude that the sample proportion differs significantly from 0.90.

Now for the P -value approach . Again, needing to allow for the possibility that the sample proportion is either too large or too small, we multiply the P -value we obtain for the one-tailed test by 2:

That is, the P -value is:

\(P=P(|Z|\geq 1.92)=P(Z>1.92 \text{ or } Z<-1.92)=2 \times 0.0274=0.055\)

Because the P -value 0.055 is (just barely) greater than the significance level \(\alpha = 0.05\), we barely fail to reject the null hypothesis. Again, we would say that there is insufficient evidence at the \(\alpha = 0.05\) level to conclude that the sample proportion differs significantly from 0.90.

Let's close this example by formalizing the definition of a P -value, as well as summarizing the P -value approach to conducting a hypothesis test.

The P -value is the smallest significance level \(\alpha\) that leads us to reject the null hypothesis.

Alternatively (and the way I prefer to think of P -values), the P -value is the probability that we'd observe a more extreme statistic than we did if the null hypothesis were true.

If the P -value is small, that is, if \(P ≤ \alpha\), then we reject the null hypothesis \(H_0\).

Note! Section  

By the way, to test \(H_0 \colon p = p_0\), some statisticians will use the test statistic:

\(Z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}}\)

rather than the one we've been using:

\(Z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}\)

One advantage of doing so is that the interpretation of the confidence interval — does it contain \(p_0\)? — is always consistent with the hypothesis test decision, as illustrated here:

For the sake of ease, let:

\(se(\hat{p})=\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\)

Two-tailed test. In this case, the critical region approach tells us to reject the null hypothesis \(H_0 \colon p = p_0\) against the alternative hypothesis \(H_A \colon p \ne p_0\):

if \(Z=\dfrac{\hat{p}-p_0}{se(\hat{p})} \geq z_{\alpha/2}\) or if \(Z=\dfrac{\hat{p}-p_0}{se(\hat{p})} \leq -z_{\alpha/2}\)

which is equivalent to rejecting the null hypothesis:

if \(\hat{p}-p_0 \geq z_{\alpha/2}se(\hat{p})\) or if \(\hat{p}-p_0 \leq -z_{\alpha/2}se(\hat{p})\)

if \(p_0 \geq \hat{p}+z_{\alpha/2}se(\hat{p})\) or if \(p_0 \leq \hat{p}-z_{\alpha/2}se(\hat{p})\)

That's the same as saying that we should reject the null hypothesis \(H_0 \text{ if } p_0\) is not in the \(\left(1-\alpha\right)100\%\) confidence interval!

Left-tailed test. In this case, the critical region approach tells us to reject the null hypothesis \(H_0 \colon p = p_0\) against the alternative hypothesis \(H_A \colon p < p_0\):

if \(Z=\dfrac{\hat{p}-p_0}{se(\hat{p})} \leq -z_{\alpha}\)

if \(\hat{p}-p_0 \leq -z_{\alpha}se(\hat{p})\)

if \(p_0 \geq \hat{p}+z_{\alpha}se(\hat{p})\)

That's the same as saying that we should reject the null hypothesis \(H_0 \text{ if } p_0\) is not in the upper \(\left(1-\alpha\right)100\%\) confidence interval:

\((0,\hat{p}+z_{\alpha}se(\hat{p}))\)

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Understanding Hypothesis Tests: Significance Levels (Alpha) and P values in Statistics

Topics: Hypothesis Testing , Statistics

What do significance levels and P values mean in hypothesis tests? What is statistical significance anyway? In this post, I’ll continue to focus on concepts and graphs to help you gain a more intuitive understanding of how hypothesis tests work in statistics.

To bring it to life, I’ll add the significance level and P value to the graph in my previous post in order to perform a graphical version of the 1 sample t-test. It’s easier to understand when you can see what statistical significance truly means!

Here’s where we left off in my last post . We want to determine whether our sample mean (330.6) indicates that this year's average energy cost is significantly different from last year’s average energy cost of $260.

The probability distribution plot above shows the distribution of sample means we’d obtain under the assumption that the null hypothesis is true (population mean = 260) and we repeatedly drew a large number of random samples.

I left you with a question: where do we draw the line for statistical significance on the graph? Now we'll add in the significance level and the P value, which are the decision-making tools we'll need.

We'll use these tools to test the following hypotheses:

• Null hypothesis: The population mean equals the hypothesized mean (260).
• Alternative hypothesis: The population mean differs from the hypothesized mean (260).

What Is the Significance Level (Alpha)?

The significance level, also denoted as alpha or α, is the probability of rejecting the null hypothesis when it is true. For example, a significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.

These types of definitions can be hard to understand because of their technical nature. A picture makes the concepts much easier to comprehend!

The significance level determines how far out from the null hypothesis value we'll draw that line on the graph. To graph a significance level of 0.05, we need to shade the 5% of the distribution that is furthest away from the null hypothesis.

In the graph above, the two shaded areas are equidistant from the null hypothesis value and each area has a probability of 0.025, for a total of 0.05. In statistics, we call these shaded areas the critical region for a two-tailed test. If the population mean is 260, we’d expect to obtain a sample mean that falls in the critical region 5% of the time. The critical region defines how far away our sample statistic must be from the null hypothesis value before we can say it is unusual enough to reject the null hypothesis.

Our sample mean (330.6) falls within the critical region, which indicates it is statistically significant at the 0.05 level.

We can also see if it is statistically significant using the other common significance level of 0.01.

The two shaded areas each have a probability of 0.005, which adds up to a total probability of 0.01. This time our sample mean does not fall within the critical region and we fail to reject the null hypothesis. This comparison shows why you need to choose your significance level before you begin your study. It protects you from choosing a significance level because it conveniently gives you significant results!

Thanks to the graph, we were able to determine that our results are statistically significant at the 0.05 level without using a P value. However, when you use the numeric output produced by statistical software , you’ll need to compare the P value to your significance level to make this determination.

Ready for a demo of Minitab Statistical Software? Just ask! 

What Are P values?

P-values are the probability of obtaining an effect at least as extreme as the one in your sample data, assuming the truth of the null hypothesis.

This definition of P values, while technically correct, is a bit convoluted. It’s easier to understand with a graph!

To graph the P value for our example data set, we need to determine the distance between the sample mean and the null hypothesis value (330.6 - 260 = 70.6). Next, we can graph the probability of obtaining a sample mean that is at least as extreme in both tails of the distribution (260 +/- 70.6).

In the graph above, the two shaded areas each have a probability of 0.01556, for a total probability 0.03112. This probability represents the likelihood of obtaining a sample mean that is at least as extreme as our sample mean in both tails of the distribution if the population mean is 260. That’s our P value!

When a P value is less than or equal to the significance level, you reject the null hypothesis. If we take the P value for our example and compare it to the common significance levels, it matches the previous graphical results. The P value of 0.03112 is statistically significant at an alpha level of 0.05, but not at the 0.01 level.

If we stick to a significance level of 0.05, we can conclude that the average energy cost for the population is greater than 260.

A common mistake is to interpret the P-value as the probability that the null hypothesis is true. To understand why this interpretation is incorrect, please read my blog post  How to Correctly Interpret P Values .

Discussion about Statistically Significant Results

A hypothesis test evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. A test result is statistically significant when the sample statistic is unusual enough relative to the null hypothesis that we can reject the null hypothesis for the entire population. “Unusual enough” in a hypothesis test is defined by:

• The assumption that the null hypothesis is true—the graphs are centered on the null hypothesis value.
• The significance level—how far out do we draw the line for the critical region?
• Our sample statistic—does it fall in the critical region?

Keep in mind that there is no magic significance level that distinguishes between the studies that have a true effect and those that don’t with 100% accuracy. The common alpha values of 0.05 and 0.01 are simply based on tradition. For a significance level of 0.05, expect to obtain sample means in the critical region 5% of the time when the null hypothesis is true . In these cases, you won’t know that the null hypothesis is true but you’ll reject it because the sample mean falls in the critical region. That’s why the significance level is also referred to as an error rate!

This type of error doesn’t imply that the experimenter did anything wrong or require any other unusual explanation. The graphs show that when the null hypothesis is true, it is possible to obtain these unusual sample means for no reason other than random sampling error. It’s just luck of the draw.

Significance levels and P values are important tools that help you quantify and control this type of error in a hypothesis test. Using these tools to decide when to reject the null hypothesis increases your chance of making the correct decision.

If you like this post, you might want to read the other posts in this series that use the same graphical framework:

• Previous: Why We Need to Use Hypothesis Tests
• Next: Confidence Intervals and Confidence Levels

If you'd like to see how I made these graphs, please read: How to Create a Graphical Version of the 1-sample t-Test .

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2 Answers 2

It will be the case that if you observed a sample that's impossible under the null (and if the statistic is able to detect that), you can get a p-value of exactly zero.

That can happen in real world problems. For example, if you do an Anderson-Darling test of goodness of fit of data to a standard uniform with some data outside that range - e.g. where your sample is (0.430, 0.712, 0.885, 1.08) - the p-value is actually zero (but a Kolmogorov-Smirnov test by contrast would give a p-value that isn't zero, even though we can rule it out by inspection).

Likelihood ratio tests will likewise give a p-value of zero if the sample is not possible under the null.

As whuber mentioned in comments, hypothesis tests don't evaluate the probability of the null hypothesis (or the alternative).

We don't (can't, really) talk about the probability of the null being true in that framework (we can do it explicitly in a Bayesian framework, though -- but then we cast the decision problem somewhat differently from the outset).

• 3 $\begingroup$ In the standard hypothesis testing framework there is no meaning to "the probability of the null hypothesis." We know that you know that but it looks like the OP doesn't. $\endgroup$ –  whuber ♦ Commented Mar 18, 2014 at 20:47
• 1 $\begingroup$ Perhaps explicating this a bit: The standard uniform includes only values from 0 to 1. Thus, a value of 1.08 is impossible. But this is really rather odd; is there a situation where we would think that a continuous variable is distributed uniformly, but not know its maximum? And if we knew its maximum was 1, then 1.08 would just be a sign of a data entry error. $\endgroup$ –  Peter Flom Commented Mar 18, 2014 at 21:05
• $\begingroup$ @whuber Does it work if I rephrase to "If so, would it mean that null hypothesis is definitely false"? $\endgroup$ –  user1205901 - Слава Україні Commented Mar 18, 2014 at 21:09
• 3 $\begingroup$ @whuber Okay, thanks, I can certainly do that, and I'll get rid of my rambling comments as well. I'm not thinking clearly this morning ... in respect of your last sentence, can you give me a hint about what sort of circumstances that comes up in? $\endgroup$ –  Glen_b Commented Mar 18, 2014 at 23:00
• 1 $\begingroup$ @whuber I'd also be interested in which circumstances a true $H_0$ can have a (true) zero p . I think that's very relevant to this question here, but it might be sufficiently different to be worth asking as a question in its own right. $\endgroup$ –  Silverfish Commented Feb 4, 2015 at 15:20

In R, the binomial test gives a P value of 'TRUE' presumably 0, if all trials succeed and hypothesis is 100% success, even if number of trials is just 1:

• $\begingroup$ That's interesting. Looking at the code, if p==1 the value computed for PVAL is (x==n) . It does a similar trick when p==0 , giving (x==0) for PVAL . $\endgroup$ –  Glen_b Commented Mar 27, 2015 at 2:04
• $\begingroup$ However, if I put in x=1,n=2,p=1 , it doesn't return FALSE , but the smallest p-value it can return, so it doesn't get to that point in the code in that case (similarly with x=1,n=1,p=0 ). So it seems as if that piece of code perhaps will only be run when it's going to return TRUE . $\endgroup$ –  Glen_b Commented Mar 27, 2015 at 2:10

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P-Value: Comprehensive Guide to Understand, Apply, and Interpret

A p-value is a statistical metric used to assess a hypothesis by comparing it with observed data.

This article delves into the concept of p-value, its calculation, interpretation, and significance. It also explores the factors that influence p-value and highlights its limitations.

Table of Content

• What is P-value?

How P-value is calculated?

How to interpret p-value, p-value in hypothesis testing, implementing p-value in python, applications of p-value, what is the p-value.

The p-value, or probability value, is a statistical measure used in hypothesis testing to assess the strength of evidence against a null hypothesis. It represents the probability of obtaining results as extreme as, or more extreme than, the observed results under the assumption that the null hypothesis is true.

In simpler words, it is used to reject or support the null hypothesis during hypothesis testing. In data science, it gives valuable insights on the statistical significance of an independent variable in predicting the dependent variable. 

Calculating the p-value typically involves the following steps:

• Formulate the Null Hypothesis (H0) : Clearly state the null hypothesis, which typically states that there is no significant relationship or effect between the variables.
• Choose an Alternative Hypothesis (H1) : Define the alternative hypothesis, which proposes the existence of a significant relationship or effect between the variables.
• Determine the Test Statistic : Calculate the test statistic, which is a measure of the discrepancy between the observed data and the expected values under the null hypothesis. The choice of test statistic depends on the type of data and the specific research question.
• Identify the Distribution of the Test Statistic : Determine the appropriate sampling distribution for the test statistic under the null hypothesis. This distribution represents the expected values of the test statistic if the null hypothesis is true.
• Calculate the Critical-value : Based on the observed test statistic and the sampling distribution, find the probability of obtaining the observed test statistic or a more extreme one, assuming the null hypothesis is true.
• Interpret the results: Compare the critical-value with t-statistic. If the t-statistic is larger than the critical value, it provides evidence to reject the null hypothesis, and vice-versa.

Its interpretation depends on the specific test and the context of the analysis. Several popular methods for calculating test statistics that are utilized in p-value calculations.

In general, a small p-value indicates that the observed data is unlikely to have occurred by random chance alone, which leads to the rejection of the null hypothesis. However, it’s crucial to choose the appropriate test based on the nature of the data and the research question, as well as to interpret the p-value in the context of the specific test being used.

The table given below shows the importance of p-value and shows the various kinds of errors that occur during hypothesis testing.

Type I error: Incorrect rejection of the null hypothesis. It is denoted by α (significance level). Type II error: Incorrect acceptance of the null hypothesis. It is denoted by β (power level)

Let’s consider an example to illustrate the process of calculating a p-value for Two Sample T-Test:

A researcher wants to investigate whether there is a significant difference in mean height between males and females in a population of university students.

Suppose we have the following data:

Starting with interpreting the process of calculating p-value

Step 1 : Formulate the Null Hypothesis (H0):

H0: There is no significant difference in mean height between males and females.

Step 2 : Choose an Alternative Hypothesis (H1):

H1: There is a significant difference in mean height between males and females.

Step 3 : Determine the Test Statistic:

The appropriate test statistic for this scenario is the two-sample t-test, which compares the means of two independent groups.

The t-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.

• s1 = First sample’s standard deviation
• s2 = Second sample’s standard deviation
• n1 = First sample’s sample size
• n2 = Second sample’s sample size

So, the calculated two-sample t-test statistic (t) is approximately 5.13.

Step 4 : Identify the Distribution of the Test Statistic:

The t-distribution is used for the two-sample t-test . The degrees of freedom for the t-distribution are determined by the sample sizes of the two groups.

 The t-distribution is a probability distribution with tails that are thicker than those of the normal distribution.

• where, n1 is total number of values for 1st category.
• n2 is total number of values for 2nd category.

The degrees of freedom (63) represent the variability available in the data to estimate the population parameters. In the context of the two-sample t-test, higher degrees of freedom provide a more precise estimate of the population variance, influencing the shape and characteristics of the t-distribution.

T-Statistic

The t-distribution is symmetric and bell-shaped, similar to the normal distribution. As the degrees of freedom increase, the t-distribution approaches the shape of the standard normal distribution. Practically, it affects the critical values used to determine statistical significance and confidence intervals.

Step 5 : Calculate Critical Value.

To find the critical t-value with a t-statistic of 5.13 and 63 degrees of freedom, we can either consult a t-table or use statistical software.

We can use scipy.stats module in Python to find the critical t-value using below code.

Comparing with T-Statistic:

The larger t-statistic suggests that the observed difference between the sample means is unlikely to have occurred by random chance alone. Therefore, we reject the null hypothesis.

• p ≤ (α = 0.05) : Reject the null hypothesis. There is sufficient evidence to conclude that the observed effect or relationship is statistically significant, meaning it is unlikely to have occurred by chance alone.
• p > (α = 0.05) : reject alternate hypothesis (or accept null hypothesis). The observed effect or relationship does not provide enough evidence to reject the null hypothesis. This does not necessarily mean there is no effect; it simply means the sample data does not provide strong enough evidence to rule out the possibility that the effect is due to chance.

In case the significance level is not specified, consider the below general inferences while interpreting your results. 

• If p > .10: not significant
• If p ≤ .10: slightly significant
• If p ≤ .05: significant
• If p ≤ .001: highly significant

Graphically, the p-value is located at the tails of any confidence interval. [As shown in fig 1]

Fig 1: Graphical Representation 

What influences p-value?

The p-value in hypothesis testing is influenced by several factors:

• Sample Size : Larger sample sizes tend to yield smaller p-values, increasing the likelihood of detecting significant effects.
• Effect Size: A larger effect size results in smaller p-values, making it easier to detect a significant relationship.
• Variability in the Data : Greater variability often leads to larger p-values, making it harder to identify significant effects.
• Significance Level : A lower chosen significance level increases the threshold for considering p-values as significant.
• Choice of Test: Different statistical tests may yield different p-values for the same data.
• Assumptions of the Test : Violations of test assumptions can impact p-values.

Understanding these factors is crucial for interpreting p-values accurately and making informed decisions in hypothesis testing.

Significance of P-value

• The p-value provides a quantitative measure of the strength of the evidence against the null hypothesis.
• Decision-Making in Hypothesis Testing
• P-value serves as a guide for interpreting the results of a statistical test. A small p-value suggests that the observed effect or relationship is statistically significant, but it does not necessarily mean that it is practically or clinically meaningful.

Limitations of P-value

• The p-value is not a direct measure of the effect size, which represents the magnitude of the observed relationship or difference between variables. A small p-value does not necessarily mean that the effect size is large or practically meaningful.
• Influenced by Various Factors

The p-value is a crucial concept in statistical hypothesis testing, serving as a guide for making decisions about the significance of the observed relationship or effect between variables.

Let’s consider a scenario where a tutor believes that the average exam score of their students is equal to the national average (85). The tutor collects a sample of exam scores from their students and performs a one-sample t-test to compare it to the population mean (85).

• The code performs a one-sample t-test to compare the mean of a sample data set to a hypothesized population mean.
• It utilizes the scipy.stats library to calculate the t-statistic and p-value. SciPy is a Python library that provides efficient numerical routines for scientific computing.
• The p-value is compared to a significance level (alpha) to determine whether to reject the null hypothesis.

Since, 0.7059>0.05 , we would conclude to fail to reject the null hypothesis. This means that, based on the sample data, there isn’t enough evidence to claim a significant difference in the exam scores of the tutor’s students compared to the national average. The tutor would accept the null hypothesis, suggesting that the average exam score of their students is statistically consistent with the national average.

• During Forward and Backward propagation: When fitting a model (say a Multiple Linear Regression model), we use the p-value in order to find the most significant variables that contribute significantly in predicting the output.
• Effects of various drug medicines: It is highly used in the field of medical research in determining whether the constituents of any drug will have the desired effect on humans or not. P-value is a very strong statistical tool used in hypothesis testing. It provides a plethora of valuable information while making an important decision like making a business intelligence inference or determining whether a drug should be used on humans or not, etc. For any doubt/query, comment below.

The p-value is a crucial concept in statistical hypothesis testing, providing a quantitative measure of the strength of evidence against the null hypothesis. It guides decision-making by comparing the p-value to a chosen significance level, typically 0.05. A small p-value indicates strong evidence against the null hypothesis, suggesting a statistically significant relationship or effect. However, the p-value is influenced by various factors and should be interpreted alongside other considerations, such as effect size and context.

Frequently Based Questions (FAQs)

Why is p-value greater than 1.

A p-value is a probability, and probabilities must be between 0 and 1. Therefore, a p-value greater than 1 is not possible.

What does P 0.01 mean?

It means that the observed test statistic is unlikely to occur by chance if the null hypothesis is true. It represents a 1% chance of observing the test statistic or a more extreme one under the null hypothesis.

Is 0.9 a good p-value?

A good p-value is typically less than or equal to 0.05, indicating that the null hypothesis is likely false and the observed relationship or effect is statistically significant.

What is p-value in a model?

It is a measure of the statistical significance of a parameter in the model. It represents the probability of obtaining the observed value of the parameter or a more extreme one, assuming the null hypothesis is true.

Why is p-value so low?

A low p-value means that the observed test statistic is unlikely to occur by chance if the null hypothesis is true. It suggests that the observed relationship or effect is statistically significant and not due to random sampling variation.

How Can You Use P-value to Compare Two Different Results of a Hypothesis Test?

Compare p-values: Lower p-value indicates stronger evidence against null hypothesis, favoring results with smaller p-values in hypothesis testing.

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Statistical tests, P values, confidence intervals, and power: a guide to misinterpretations

Sander greenland.

Department of Epidemiology and Department of Statistics, University of California, Los Angeles, CA USA

Stephen J. Senn

Competence Center for Methodology and Statistics, Luxembourg Institute of Health, Strassen, Luxembourg

Kenneth J. Rothman

RTI Health Solutions, Research Triangle Institute, Research Triangle Park, NC USA

John B. Carlin

Clinical Epidemiology and Biostatistics Unit, Murdoch Children’s Research Institute, School of Population Health, University of Melbourne, Melbourne, VIC Australia

Charles Poole

Department of Epidemiology, Gillings School of Global Public Health, University of North Carolina, Chapel Hill, NC USA

Steven N. Goodman

Meta-Research Innovation Center, Departments of Medicine and of Health Research and Policy, Stanford University School of Medicine, Stanford, CA USA

Douglas G. Altman

Centre for Statistics in Medicine, Nuffield Department of Orthopaedics, Rheumatology and Musculoskeletal Sciences, University of Oxford, Oxford, UK

Misinterpretation and abuse of statistical tests, confidence intervals, and statistical power have been decried for decades, yet remain rampant. A key problem is that there are no interpretations of these concepts that are at once simple, intuitive, correct, and foolproof. Instead, correct use and interpretation of these statistics requires an attention to detail which seems to tax the patience of working scientists. This high cognitive demand has led to an epidemic of shortcut definitions and interpretations that are simply wrong, sometimes disastrously so—and yet these misinterpretations dominate much of the scientific literature. In light of this problem, we provide definitions and a discussion of basic statistics that are more general and critical than typically found in traditional introductory expositions. Our goal is to provide a resource for instructors, researchers, and consumers of statistics whose knowledge of statistical theory and technique may be limited but who wish to avoid and spot misinterpretations. We emphasize how violation of often unstated analysis protocols (such as selecting analyses for presentation based on the P values they produce) can lead to small P values even if the declared test hypothesis is correct, and can lead to large P values even if that hypothesis is incorrect. We then provide an explanatory list of 25 misinterpretations of P values, confidence intervals, and power. We conclude with guidelines for improving statistical interpretation and reporting.

Introduction

Misinterpretation and abuse of statistical tests has been decried for decades, yet remains so rampant that some scientific journals discourage use of “statistical significance” (classifying results as “significant” or not based on a P value) [ 1 ]. One journal now bans all statistical tests and mathematically related procedures such as confidence intervals [ 2 ], which has led to considerable discussion and debate about the merits of such bans [ 3 , 4 ].

Despite such bans, we expect that the statistical methods at issue will be with us for many years to come. We thus think it imperative that basic teaching as well as general understanding of these methods be improved. Toward that end, we attempt to explain the meaning of significance tests, confidence intervals, and statistical power in a more general and critical way than is traditionally done, and then review 25 common misconceptions in light of our explanations. We also discuss a few more subtle but nonetheless pervasive problems, explaining why it is important to examine and synthesize all results relating to a scientific question, rather than focus on individual findings. We further explain why statistical tests should never constitute the sole input to inferences or decisions about associations or effects. Among the many reasons are that, in most scientific settings, the arbitrary classification of results into “significant” and “non-significant” is unnecessary for and often damaging to valid interpretation of data; and that estimation of the size of effects and the uncertainty surrounding our estimates will be far more important for scientific inference and sound judgment than any such classification.

More detailed discussion of the general issues can be found in many articles, chapters, and books on statistical methods and their interpretation [ 5 – 20 ]. Specific issues are covered at length in these sources and in the many peer-reviewed articles that critique common misinterpretations of null-hypothesis testing and “statistical significance” [ 1 , 12 , 21 – 74 ].

Statistical tests, P values, and confidence intervals: a caustic primer

Statistical models, hypotheses, and tests.

Every method of statistical inference depends on a complex web of assumptions about how data were collected and analyzed, and how the analysis results were selected for presentation. The full set of assumptions is embodied in a statistical model that underpins the method. This model is a mathematical representation of data variability, and thus ideally would capture accurately all sources of such variability. Many problems arise however because this statistical model often incorporates unrealistic or at best unjustified assumptions. This is true even for so-called “non-parametric” methods, which (like other methods) depend on assumptions of random sampling or randomization. These assumptions are often deceptively simple to write down mathematically, yet in practice are difficult to satisfy and verify, as they may depend on successful completion of a long sequence of actions (such as identifying, contacting, obtaining consent from, obtaining cooperation of, and following up subjects, as well as adherence to study protocols for treatment allocation, masking, and data analysis).

There is also a serious problem of defining the scope of a model, in that it should allow not only for a good representation of the observed data but also of hypothetical alternative data that might have been observed. The reference frame for data that “might have been observed” is often unclear, for example if multiple outcome measures or multiple predictive factors have been measured, and many decisions surrounding analysis choices have been made after the data were collected—as is invariably the case [ 33 ].

The difficulty of understanding and assessing underlying assumptions is exacerbated by the fact that the statistical model is usually presented in a highly compressed and abstract form—if presented at all. As a result, many assumptions go unremarked and are often unrecognized by users as well as consumers of statistics. Nonetheless, all statistical methods and interpretations are premised on the model assumptions; that is, on an assumption that the model provides a valid representation of the variation we would expect to see across data sets, faithfully reflecting the circumstances surrounding the study and phenomena occurring within it.

In most applications of statistical testing, one assumption in the model is a hypothesis that a particular effect has a specific size, and has been targeted for statistical analysis. (For simplicity, we use the word “effect” when “association or effect” would arguably be better in allowing for noncausal studies such as most surveys.) This targeted assumption is called the study hypothesis or test hypothesis , and the statistical methods used to evaluate it are called statistical hypothesis tests . Most often, the targeted effect size is a “null” value representing zero effect (e.g., that the study treatment makes no difference in average outcome), in which case the test hypothesis is called the null hypothesis . Nonetheless, it is also possible to test other effect sizes. We may also test hypotheses that the effect does or does not fall within a specific range; for example, we may test the hypothesis that the effect is no greater than a particular amount, in which case the hypothesis is said to be a one - sided or dividing hypothesis [ 7 , 8 ].

Much statistical teaching and practice has developed a strong (and unhealthy) focus on the idea that the main aim of a study should be to test null hypotheses. In fact most descriptions of statistical testing focus only on testing null hypotheses, and the entire topic has been called “Null Hypothesis Significance Testing” (NHST). This exclusive focus on null hypotheses contributes to misunderstanding of tests. Adding to the misunderstanding is that many authors (including R.A. Fisher) use “null hypothesis” to refer to any test hypothesis, even though this usage is at odds with other authors and with ordinary English definitions of “null”—as are statistical usages of “significance” and “confidence.”

Uncertainty, probability, and statistical significance

A more refined goal of statistical analysis is to provide an evaluation of certainty or uncertainty regarding the size of an effect. It is natural to express such certainty in terms of “probabilities” of hypotheses. In conventional statistical methods, however, “probability” refers not to hypotheses, but to quantities that are hypothetical frequencies of data patterns under an assumed statistical model. These methods are thus called frequentist methods, and the hypothetical frequencies they predict are called “frequency probabilities.” Despite considerable training to the contrary, many statistically educated scientists revert to the habit of misinterpreting these frequency probabilities as hypothesis probabilities. (Even more confusingly, the term “likelihood of a parameter value” is reserved by statisticians to refer to the probability of the observed data given the parameter value; it does not refer to a probability of the parameter taking on the given value.)

Nowhere are these problems more rampant than in applications of a hypothetical frequency called the P value, also known as the “observed significance level” for the test hypothesis. Statistical “significance tests” based on this concept have been a central part of statistical analyses for centuries [ 75 ]. The focus of traditional definitions of P values and statistical significance has been on null hypotheses, treating all other assumptions used to compute the P value as if they were known to be correct. Recognizing that these other assumptions are often questionable if not unwarranted, we will adopt a more general view of the P value as a statistical summary of the compatibility between the observed data and what we would predict or expect to see if we knew the entire statistical model ( all the assumptions used to compute the P value) were correct.

Specifically, the distance between the data and the model prediction is measured using a test statistic (such as a t-statistic or a Chi squared statistic). The P value is then the probability that the chosen test statistic would have been at least as large as its observed value if every model assumption were correct, including the test hypothesis. This definition embodies a crucial point lost in traditional definitions: In logical terms, the P value tests all the assumptions about how the data were generated (the entire model), not just the targeted hypothesis it is supposed to test (such as a null hypothesis). Furthermore, these assumptions include far more than what are traditionally presented as modeling or probability assumptions—they include assumptions about the conduct of the analysis, for example that intermediate analysis results were not used to determine which analyses would be presented.

It is true that the smaller the P value, the more unusual the data would be if every single assumption were correct; but a very small P value does not tell us which assumption is incorrect. For example, the P value may be very small because the targeted hypothesis is false; but it may instead (or in addition) be very small because the study protocols were violated, or because it was selected for presentation based on its small size. Conversely, a large P value indicates only that the data are not unusual under the model, but does not imply that the model or any aspect of it (such as the targeted hypothesis) is correct; it may instead (or in addition) be large because (again) the study protocols were violated, or because it was selected for presentation based on its large size.

The general definition of a P value may help one to understand why statistical tests tell us much less than what many think they do: Not only does a P value not tell us whether the hypothesis targeted for testing is true or not; it says nothing specifically related to that hypothesis unless we can be completely assured that every other assumption used for its computation is correct—an assurance that is lacking in far too many studies.

Nonetheless, the P value can be viewed as a continuous measure of the compatibility between the data and the entire model used to compute it, ranging from 0 for complete incompatibility to 1 for perfect compatibility, and in this sense may be viewed as measuring the fit of the model to the data. Too often, however, the P value is degraded into a dichotomy in which results are declared “statistically significant” if P falls on or below a cut-off (usually 0.05) and declared “nonsignificant” otherwise. The terms “significance level” and “alpha level” (α) are often used to refer to the cut-off; however, the term “significance level” invites confusion of the cut-off with the P value itself. Their difference is profound: the cut-off value α is supposed to be fixed in advance and is thus part of the study design, unchanged in light of the data. In contrast, the P value is a number computed from the data and thus an analysis result, unknown until it is computed.

Moving from tests to estimates

We can vary the test hypothesis while leaving other assumptions unchanged, to see how the P value differs across competing test hypotheses. Usually, these test hypotheses specify different sizes for a targeted effect; for example, we may test the hypothesis that the average difference between two treatment groups is zero (the null hypothesis), or that it is 20 or −10 or any size of interest. The effect size whose test produced P  = 1 is the size most compatible with the data (in the sense of predicting what was in fact observed) if all the other assumptions used in the test (the statistical model) were correct, and provides a point estimate of the effect under those assumptions. The effect sizes whose test produced P  > 0.05 will typically define a range of sizes (e.g., from 11.0 to 19.5) that would be considered more compatible with the data (in the sense of the observations being closer to what the model predicted) than sizes outside the range—again, if the statistical model were correct. This range corresponds to a 1 − 0.05 = 0.95 or 95 % confidence interval , and provides a convenient way of summarizing the results of hypothesis tests for many effect sizes. Confidence intervals are examples of interval estimates .

Neyman [ 76 ] proposed the construction of confidence intervals in this way because they have the following property: If one calculates, say, 95 % confidence intervals repeatedly in valid applications , 95 % of them, on average, will contain (i.e., include or cover) the true effect size. Hence, the specified confidence level is called the coverage probability. As Neyman stressed repeatedly, this coverage probability is a property of a long sequence of confidence intervals computed from valid models, rather than a property of any single confidence interval.

Many journals now require confidence intervals, but most textbooks and studies discuss P values only for the null hypothesis of no effect. This exclusive focus on null hypotheses in testing not only contributes to misunderstanding of tests and underappreciation of estimation, but also obscures the close relationship between P values and confidence intervals, as well as the weaknesses they share.

What P values, confidence intervals, and power calculations don’t tell us

Much distortion arises from basic misunderstanding of what P values and their relatives (such as confidence intervals) do not tell us. Therefore, based on the articles in our reference list, we review prevalent P value misinterpretations as a way of moving toward defensible interpretations and presentations. We adopt the format of Goodman [ 40 ] in providing a list of misinterpretations that can be used to critically evaluate conclusions offered by research reports and reviews. Every one of the bolded statements in our list has contributed to statistical distortion of the scientific literature, and we add the emphatic “No!” to underscore statements that are not only fallacious but also not “true enough for practical purposes.”

Common misinterpretations of single P values

• The P value is the probability that the test hypothesis is true; for example, if a test of the null hypothesis gave P   =   0.01, the null hypothesis has only a 1   % chance of being true; if instead it gave P   =   0.40, the null hypothesis has a 40   % chance of being true . No! The P value assumes the test hypothesis is true—it is not a hypothesis probability and may be far from any reasonable probability for the test hypothesis. The P value simply indicates the degree to which the data conform to the pattern predicted by the test hypothesis and all the other assumptions used in the test (the underlying statistical model). Thus P  = 0.01 would indicate that the data are not very close to what the statistical model (including the test hypothesis) predicted they should be, while P  = 0.40 would indicate that the data are much closer to the model prediction, allowing for chance variation.

Note : One often sees “alone” dropped from this description (becoming “the P value for the null hypothesis is the probability that chance produced the observed association”), so that the statement is more ambiguous, but just as wrong.

• A significant test result ( P   ≤   0.05) means that the test hypothesis is false or should be rejected . No! A small P value simply flags the data as being unusual if all the assumptions used to compute it (including the test hypothesis) were correct; it may be small because there was a large random error or because some assumption other than the test hypothesis was violated (for example, the assumption that this P value was not selected for presentation because it was below 0.05). P  ≤ 0.05 only means that a discrepancy from the hypothesis prediction (e.g., no difference between treatment groups) would be as large or larger than that observed no more than 5 % of the time if only chance were creating the discrepancy (as opposed to a violation of the test hypothesis or a mistaken assumption).
• A nonsignificant test result ( P   >   0.05) means that the test hypothesis is true or should be accepted . No! A large P value only suggests that the data are not unusual if all the assumptions used to compute the P value (including the test hypothesis) were correct. The same data would also not be unusual under many other hypotheses. Furthermore, even if the test hypothesis is wrong, the P value may be large because it was inflated by a large random error or because of some other erroneous assumption (for example, the assumption that this P value was not selected for presentation because it was above 0.05). P  > 0.05 only means that a discrepancy from the hypothesis prediction (e.g., no difference between treatment groups) would be as large or larger than that observed more than 5 % of the time if only chance were creating the discrepancy.
• A large P value is evidence in favor of the test hypothesis . No! In fact, any P value less than 1 implies that the test hypothesis is not the hypothesis most compatible with the data, because any other hypothesis with a larger P value would be even more compatible with the data. A P value cannot be said to favor the test hypothesis except in relation to those hypotheses with smaller P values. Furthermore, a large P value often indicates only that the data are incapable of discriminating among many competing hypotheses (as would be seen immediately by examining the range of the confidence interval). For example, many authors will misinterpret P  = 0.70 from a test of the null hypothesis as evidence for no effect, when in fact it indicates that, even though the null hypothesis is compatible with the data under the assumptions used to compute the P value, it is not the hypothesis most compatible with the data—that honor would belong to a hypothesis with P  = 1. But even if P  = 1, there will be many other hypotheses that are highly consistent with the data, so that a definitive conclusion of “no association” cannot be deduced from a P value, no matter how large.
• A null -hypothesis P value greater than 0.05 means that no effect was observed, or that absence of an effect was shown or demonstrated . No! Observing P  > 0.05 for the null hypothesis only means that the null is one among the many hypotheses that have P  > 0.05. Thus, unless the point estimate (observed association) equals the null value exactly, it is a mistake to conclude from P  > 0.05 that a study found “no association” or “no evidence” of an effect. If the null P value is less than 1 some association must be present in the data, and one must look at the point estimate to determine the effect size most compatible with the data under the assumed model.
• Statistical significance indicates a scientifically or substantively important relation has been detected . No! Especially when a study is large, very minor effects or small assumption violations can lead to statistically significant tests of the null hypothesis. Again, a small null P value simply flags the data as being unusual if all the assumptions used to compute it (including the null hypothesis) were correct; but the way the data are unusual might be of no clinical interest. One must look at the confidence interval to determine which effect sizes of scientific or other substantive (e.g., clinical) importance are relatively compatible with the data, given the model.
• Lack of statistical significance indicates that the effect size is small . No! Especially when a study is small, even large effects may be “drowned in noise” and thus fail to be detected as statistically significant by a statistical test. A large null P value simply flags the data as not being unusual if all the assumptions used to compute it (including the test hypothesis) were correct; but the same data will also not be unusual under many other models and hypotheses besides the null. Again, one must look at the confidence interval to determine whether it includes effect sizes of importance.
• The P value is the chance of our data occurring if the test hypothesis is true; for example, P   =   0.05 means that the observed association would occur only 5   % of the time under the test hypothesis . No! The P value refers not only to what we observed, but also observations more extreme than what we observed (where “extremity” is measured in a particular way). And again, the P value refers to a data frequency when all the assumptions used to compute it are correct. In addition to the test hypothesis, these assumptions include randomness in sampling, treatment assignment, loss, and missingness, as well as an assumption that the P value was not selected for presentation based on its size or some other aspect of the results.
• If you reject the test hypothesis because P   ≤   0.05, the chance you are in error (the chance your “significant finding” is a false positive) is 5   % . No! To see why this description is false, suppose the test hypothesis is in fact true. Then, if you reject it, the chance you are in error is 100 %, not 5 %. The 5 % refers only to how often you would reject it, and therefore be in error, over very many uses of the test across different studies when the test hypothesis and all other assumptions used for the test are true. It does not refer to your single use of the test, which may have been thrown off by assumption violations as well as random errors. This is yet another version of misinterpretation #1.
• P   =   0.05 and P   ≤   0.05 mean the same thing . No! This is like saying reported height = 2 m and reported height ≤2 m are the same thing: “height = 2 m” would include few people and those people would be considered tall, whereas “height ≤2 m” would include most people including small children. Similarly, P  = 0.05 would be considered a borderline result in terms of statistical significance, whereas P  ≤ 0.05 lumps borderline results together with results very incompatible with the model (e.g., P  = 0.0001) thus rendering its meaning vague, for no good purpose.
• P values are properly reported as inequalities (e.g., report “ P   <   0.02” when P   =   0.015 or report “ P   >   0.05” when P   =   0.06 or P   =   0.70) . No! This is bad practice because it makes it difficult or impossible for the reader to accurately interpret the statistical result. Only when the P value is very small (e.g., under 0.001) does an inequality become justifiable: There is little practical difference among very small P values when the assumptions used to compute P values are not known with enough certainty to justify such precision, and most methods for computing P values are not numerically accurate below a certain point.
• Statistical significance is a property of the phenomenon being studied, and thus statistical tests detect significance . No! This misinterpretation is promoted when researchers state that they have or have not found “evidence of” a statistically significant effect. The effect being tested either exists or does not exist. “Statistical significance” is a dichotomous description of a P value (that it is below the chosen cut-off) and thus is a property of a result of a statistical test; it is not a property of the effect or population being studied.
• One should always use two-sided P values . No! Two-sided P values are designed to test hypotheses that the targeted effect measure equals a specific value (e.g., zero), and is neither above nor below this value. When, however, the test hypothesis of scientific or practical interest is a one-sided (dividing) hypothesis, a one-sided P value is appropriate. For example, consider the practical question of whether a new drug is at least as good as the standard drug for increasing survival time. This question is one-sided, so testing this hypothesis calls for a one-sided P value. Nonetheless, because two-sided P values are the usual default, it will be important to note when and why a one-sided P value is being used instead.

There are other interpretations of P values that are controversial, in that whether a categorical “No!” is warranted depends on one’s philosophy of statistics and the precise meaning given to the terms involved. The disputed claims deserve recognition if one wishes to avoid such controversy.

For example, it has been argued that P values overstate evidence against test hypotheses, based on directly comparing P values against certain quantities (likelihood ratios and Bayes factors) that play a central role as evidence measures in Bayesian analysis [ 37 , 72 , 77 – 83 ]. Nonetheless, many other statisticians do not accept these quantities as gold standards, and instead point out that P values summarize crucial evidence needed to gauge the error rates of decisions based on statistical tests (even though they are far from sufficient for making those decisions). Thus, from this frequentist perspective, P values do not overstate evidence and may even be considered as measuring one aspect of evidence [ 7 , 8 , 84 – 87 ], with 1 −  P measuring evidence against the model used to compute the P value. See also Murtaugh [ 88 ] and its accompanying discussion.

Common misinterpretations of P value comparisons and predictions

Some of the most severe distortions of the scientific literature produced by statistical testing involve erroneous comparison and synthesis of results from different studies or study subgroups. Among the worst are:

• 15. When the same hypothesis is tested in different studies and none or a minority of the tests are statistically significant (all P   >   0.05), the overall evidence supports the hypothesis . No! This belief is often used to claim that a literature supports no effect when the opposite is case. It reflects a tendency of researchers to “overestimate the power of most research” [ 89 ]. In reality, every study could fail to reach statistical significance and yet when combined show a statistically significant association and persuasive evidence of an effect. For example, if there were five studies each with P  = 0.10, none would be significant at 0.05 level; but when these P values are combined using the Fisher formula [ 9 ], the overall P value would be 0.01. There are many real examples of persuasive evidence for important effects when few studies or even no study reported “statistically significant” associations [ 90 , 91 ]. Thus, lack of statistical significance of individual studies should not be taken as implying that the totality of evidence supports no effect.
• 16. When the same hypothesis is tested in two different populations and the resulting P values are on opposite sides of 0.05, the results are conflicting . No! Statistical tests are sensitive to many differences between study populations that are irrelevant to whether their results are in agreement, such as the sizes of compared groups in each population. As a consequence, two studies may provide very different P values for the same test hypothesis and yet be in perfect agreement (e.g., may show identical observed associations). For example, suppose we had two randomized trials A and B of a treatment, identical except that trial A had a known standard error of 2 for the mean difference between treatment groups whereas trial B had a known standard error of 1 for the difference. If both trials observed a difference between treatment groups of exactly 3, the usual normal test would produce P  = 0.13 in A but P  = 0.003 in B. Despite their difference in P values, the test of the hypothesis of no difference in effect across studies would have P  = 1, reflecting the perfect agreement of the observed mean differences from the studies. Differences between results must be evaluated by directly, for example by estimating and testing those differences to produce a confidence interval and a P value comparing the results (often called analysis of heterogeneity, interaction, or modification).
• 17. When the same hypothesis is tested in two different populations and the same P values are obtained, the results are in agreement . No! Again, tests are sensitive to many differences between populations that are irrelevant to whether their results are in agreement. Two different studies may even exhibit identical P values for testing the same hypothesis yet also exhibit clearly different observed associations. For example, suppose randomized experiment A observed a mean difference between treatment groups of 3.00 with standard error 1.00, while B observed a mean difference of 12.00 with standard error 4.00. Then the standard normal test would produce P  = 0.003 in both; yet the test of the hypothesis of no difference in effect across studies gives P  = 0.03, reflecting the large difference (12.00 − 3.00 = 9.00) between the mean differences.
• 18. If one observes a small P value, there is a good chance that the next study will produce a P value at least as small for the same hypothesis. No! This is false even under the ideal condition that both studies are independent and all assumptions including the test hypothesis are correct in both studies. In that case, if (say) one observes P  = 0.03, the chance that the new study will show P  ≤ 0.03 is only 3 %; thus the chance the new study will show a P value as small or smaller (the “replication probability”) is exactly the observed P value! If on the other hand the small P value arose solely because the true effect exactly equaled its observed estimate, there would be a 50 % chance that a repeat experiment of identical design would have a larger P value [ 37 ]. In general, the size of the new P value will be extremely sensitive to the study size and the extent to which the test hypothesis or other assumptions are violated in the new study [ 86 ]; in particular, P may be very small or very large depending on whether the study and the violations are large or small.

Finally, although it is (we hope obviously) wrong to do so, one sometimes sees the null hypothesis compared with another (alternative) hypothesis using a two-sided P value for the null and a one-sided P value for the alternative. This comparison is biased in favor of the null in that the two-sided test will falsely reject the null only half as often as the one-sided test will falsely reject the alternative (again, under all the assumptions used for testing).

Common misinterpretations of confidence intervals

Most of the above misinterpretations translate into an analogous misinterpretation for confidence intervals. For example, another misinterpretation of P  > 0.05 is that it means the test hypothesis has only a 5 % chance of being false, which in terms of a confidence interval becomes the common fallacy:

• 19. The specific 95   % confidence interval presented by a study has a 95   % chance of containing the true effect size . No! A reported confidence interval is a range between two numbers. The frequency with which an observed interval (e.g., 0.72–2.88) contains the true effect is either 100 % if the true effect is within the interval or 0 % if not; the 95 % refers only to how often 95 % confidence intervals computed from very many studies would contain the true size if all the assumptions used to compute the intervals were correct . It is possible to compute an interval that can be interpreted as having 95 % probability of containing the true value; nonetheless, such computations require not only the assumptions used to compute the confidence interval, but also further assumptions about the size of effects in the model. These further assumptions are summarized in what is called a prior distribution , and the resulting intervals are usually called Bayesian posterior (or credible) intervals to distinguish them from confidence intervals [ 18 ].

Symmetrically, the misinterpretation of a small P value as disproving the test hypothesis could be translated into:

• 20. An effect size outside the 95   % confidence interval has been refuted (or excluded) by the data . No! As with the P value, the confidence interval is computed from many assumptions, the violation of which may have led to the results. Thus it is the combination of the data with the assumptions, along with the arbitrary 95 % criterion, that are needed to declare an effect size outside the interval is in some way incompatible with the observations. Even then, judgements as extreme as saying the effect size has been refuted or excluded will require even stronger conditions.

As with P values, naïve comparison of confidence intervals can be highly misleading:

• 21. If two confidence intervals overlap, the difference between two estimates or studies is not significant . No! The 95 % confidence intervals from two subgroups or studies may overlap substantially and yet the test for difference between them may still produce P  < 0.05. Suppose for example, two 95 % confidence intervals for means from normal populations with known variances are (1.04, 4.96) and (4.16, 19.84); these intervals overlap, yet the test of the hypothesis of no difference in effect across studies gives P  = 0.03. As with P values, comparison between groups requires statistics that directly test and estimate the differences across groups. It can, however, be noted that if the two 95 % confidence intervals fail to overlap, then when using the same assumptions used to compute the confidence intervals we will find P  < 0.05 for the difference; and if one of the 95 % intervals contains the point estimate from the other group or study, we will find P  > 0.05 for the difference.

Finally, as with P values, the replication properties of confidence intervals are usually misunderstood:

• 22. An observed 95   % confidence interval predicts that 95   % of the estimates from future studies will fall inside the observed interval. No! This statement is wrong in several ways. Most importantly, under the model, 95 % is the frequency with which other unobserved intervals will contain the true effect , not how frequently the one interval being presented will contain future estimates. In fact, even under ideal conditions the chance that a future estimate will fall within the current interval will usually be much less than 95 %. For example, if two independent studies of the same quantity provide unbiased normal point estimates with the same standard errors, the chance that the 95 % confidence interval for the first study contains the point estimate from the second is 83 % (which is the chance that the difference between the two estimates is less than 1.96 standard errors). Again, an observed interval either does or does not contain the true effect; the 95 % refers only to how often 95 % confidence intervals computed from very many studies would contain the true effect if all the assumptions used to compute the intervals were correct.
• 23. If one 95   % confidence interval includes the null value and another excludes that value, the interval excluding the null is the more precise one . No! When the model is correct, precision of statistical estimation is measured directly by confidence interval width (measured on the appropriate scale). It is not a matter of inclusion or exclusion of the null or any other value. Consider two 95 % confidence intervals for a difference in means, one with limits of 5 and 40, the other with limits of −5 and 10. The first interval excludes the null value of 0, but is 30 units wide. The second includes the null value, but is half as wide and therefore much more precise.

In addition to the above misinterpretations, 95 % confidence intervals force the 0.05-level cutoff on the reader, lumping together all effect sizes with P  > 0.05, and in this way are as bad as presenting P values as dichotomies. Nonetheless, many authors agree that confidence intervals are superior to tests and P values because they allow one to shift focus away from the null hypothesis, toward the full range of effect sizes compatible with the data—a shift recommended by many authors and a growing number of journals. Another way to bring attention to non-null hypotheses is to present their P values; for example, one could provide or demand P values for those effect sizes that are recognized as scientifically reasonable alternatives to the null.

As with P values, further cautions are needed to avoid misinterpreting confidence intervals as providing sharp answers when none are warranted. The hypothesis which says the point estimate is the correct effect will have the largest P value ( P  = 1 in most cases), and hypotheses inside a confidence interval will have higher P values than hypotheses outside the interval. The P values will vary greatly, however, among hypotheses inside the interval, as well as among hypotheses on the outside. Also, two hypotheses may have nearly equal P values even though one of the hypotheses is inside the interval and the other is outside. Thus, if we use P values to measure compatibility of hypotheses with data and wish to compare hypotheses with this measure, we need to examine their P values directly, not simply ask whether the hypotheses are inside or outside the interval. This need is particularly acute when (as usual) one of the hypotheses under scrutiny is a null hypothesis.

Common misinterpretations of power

The power of a test to detect a correct alternative hypothesis is the pre-study probability that the test will reject the test hypothesis (e.g., the probability that P will not exceed a pre-specified cut-off such as 0.05). (The corresponding pre-study probability of failing to reject the test hypothesis when the alternative is correct is one minus the power, also known as the Type-II or beta error rate) [ 84 ] As with P values and confidence intervals, this probability is defined over repetitions of the same study design and so is a frequency probability. One source of reasonable alternative hypotheses are the effect sizes that were used to compute power in the study proposal. Pre-study power calculations do not, however, measure the compatibility of these alternatives with the data actually observed, while power calculated from the observed data is a direct (if obscure) transformation of the null P value and so provides no test of the alternatives. Thus, presentation of power does not obviate the need to provide interval estimates and direct tests of the alternatives.

For these reasons, many authors have condemned use of power to interpret estimates and statistical tests [ 42 , 92 – 97 ], arguing that (in contrast to confidence intervals) it distracts attention from direct comparisons of hypotheses and introduces new misinterpretations, such as:

• 24. If you accept the null hypothesis because the null P value exceeds 0.05 and the power of your test is 90   %, the chance you are in error (the chance that your finding is a false negative) is 10   % . No! If the null hypothesis is false and you accept it, the chance you are in error is 100 %, not 10 %. Conversely, if the null hypothesis is true and you accept it, the chance you are in error is 0 %. The 10 % refers only to how often you would be in error over very many uses of the test across different studies when the particular alternative used to compute power is correct and all other assumptions used for the test are correct in all the studies. It does not refer to your single use of the test or your error rate under any alternative effect size other than the one used to compute power.

It can be especially misleading to compare results for two hypotheses by presenting a test or P value for one and power for the other. For example, testing the null by seeing whether P  ≤ 0.05 with a power less than 1 − 0.05 = 0.95 for the alternative (as done routinely) will bias the comparison in favor of the null because it entails a lower probability of incorrectly rejecting the null (0.05) than of incorrectly accepting the null when the alternative is correct. Thus, claims about relative support or evidence need to be based on direct and comparable measures of support or evidence for both hypotheses, otherwise mistakes like the following will occur:

• 25. If the null P value exceeds 0.05 and the power of this test is 90   % at an alternative, the results support the null over the alternative . This claim seems intuitive to many, but counterexamples are easy to construct in which the null P value is between 0.05 and 0.10, and yet there are alternatives whose own P value exceeds 0.10 and for which the power is 0.90. Parallel results ensue for other accepted measures of compatibility, evidence, and support, indicating that the data show lower compatibility with and more evidence against the null than the alternative, despite the fact that the null P value is “not significant” at the 0.05 alpha level and the power against the alternative is “very high” [ 42 ].

Despite its shortcomings for interpreting current data, power can be useful for designing studies and for understanding why replication of “statistical significance” will often fail even under ideal conditions. Studies are often designed or claimed to have 80 % power against a key alternative when using a 0.05 significance level, although in execution often have less power due to unanticipated problems such as low subject recruitment. Thus, if the alternative is correct and the actual power of two studies is 80 %, the chance that the studies will both show P  ≤ 0.05 will at best be only 0.80(0.80) = 64 %; furthermore, the chance that one study shows P  ≤ 0.05 and the other does not (and thus will be misinterpreted as showing conflicting results) is 2(0.80)0.20 = 32 % or about 1 chance in 3. Similar calculations taking account of typical problems suggest that one could anticipate a “replication crisis” even if there were no publication or reporting bias, simply because current design and testing conventions treat individual study results as dichotomous outputs of “significant”/“nonsignificant” or “reject”/“accept.”

A statistical model is much more than an equation with Greek letters

The above list could be expanded by reviewing the research literature. We will however now turn to direct discussion of an issue that has been receiving more attention of late, yet is still widely overlooked or interpreted too narrowly in statistical teaching and presentations: That the statistical model used to obtain the results is correct.

Too often, the full statistical model is treated as a simple regression or structural equation in which effects are represented by parameters denoted by Greek letters. “Model checking” is then limited to tests of fit or testing additional terms for the model. Yet these tests of fit themselves make further assumptions that should be seen as part of the full model. For example, all common tests and confidence intervals depend on assumptions of random selection for observation or treatment and random loss or missingness within levels of controlled covariates. These assumptions have gradually come under scrutiny via sensitivity and bias analysis [ 98 ], but such methods remain far removed from the basic statistical training given to most researchers.

Less often stated is the even more crucial assumption that the analyses themselves were not guided toward finding nonsignificance or significance (analysis bias), and that the analysis results were not reported based on their nonsignificance or significance (reporting bias and publication bias). Selective reporting renders false even the limited ideal meanings of statistical significance, P values, and confidence intervals. Because author decisions to report and editorial decisions to publish results often depend on whether the P value is above or below 0.05, selective reporting has been identified as a major problem in large segments of the scientific literature [ 99 – 101 ].

Although this selection problem has also been subject to sensitivity analysis, there has been a bias in studies of reporting and publication bias: It is usually assumed that these biases favor significance. This assumption is of course correct when (as is often the case) researchers select results for presentation when P  ≤ 0.05, a practice that tends to exaggerate associations [ 101 – 105 ]. Nonetheless, bias in favor of reporting P  ≤ 0.05 is not always plausible let alone supported by evidence or common sense. For example, one might expect selection for P  > 0.05 in publications funded by those with stakes in acceptance of the null hypothesis (a practice which tends to understate associations); in accord with that expectation, some empirical studies have observed smaller estimates and “nonsignificance” more often in such publications than in other studies [ 101 , 106 , 107 ].

Addressing such problems would require far more political will and effort than addressing misinterpretation of statistics, such as enforcing registration of trials, along with open data and analysis code from all completed studies (as in the AllTrials initiative, http://www.alltrials.net/ ). In the meantime, readers are advised to consider the entire context in which research reports are produced and appear when interpreting the statistics and conclusions offered by the reports.

Conclusions

Upon realizing that statistical tests are usually misinterpreted, one may wonder what if anything these tests do for science. They were originally intended to account for random variability as a source of error, thereby sounding a note of caution against overinterpretation of observed associations as true effects or as stronger evidence against null hypotheses than was warranted. But before long that use was turned on its head to provide fallacious support for null hypotheses in the form of “failure to achieve” or “failure to attain” statistical significance.

We have no doubt that the founders of modern statistical testing would be horrified by common treatments of their invention. In their first paper describing their binary approach to statistical testing, Neyman and Pearson [ 108 ] wrote that “it is doubtful whether the knowledge that [a P value] was really 0.03 (or 0.06), rather than 0.05…would in fact ever modify our judgment” and that “The tests themselves give no final verdict, but as tools help the worker who is using them to form his final decision.” Pearson [ 109 ] later added, “No doubt we could more aptly have said, ‘his final or provisional decision.’” Fisher [ 110 ] went further, saying “No scientific worker has a fixed level of significance at which from year to year, and in all circumstances, he rejects hypotheses; he rather gives his mind to each particular case in the light of his evidence and his ideas.” Yet fallacious and ritualistic use of tests continued to spread, including beliefs that whether P was above or below 0.05 was a universal arbiter of discovery. Thus by 1965, Hill [ 111 ] lamented that “too often we weaken our capacity to interpret data and to take reasonable decisions whatever the value of P . And far too often we deduce ‘no difference’ from ‘no significant difference.’”

In response, it has been argued that some misinterpretations are harmless in tightly controlled experiments on well-understood systems, where the test hypothesis may have special support from established theories (e.g., Mendelian genetics) and in which every other assumption (such as random allocation) is forced to hold by careful design and execution of the study. But it has long been asserted that the harms of statistical testing in more uncontrollable and amorphous research settings (such as social-science, health, and medical fields) have far outweighed its benefits, leading to calls for banning such tests in research reports—again with one journal banning P values as well as confidence intervals [ 2 ].

Given, however, the deep entrenchment of statistical testing, as well as the absence of generally accepted alternative methods, there have been many attempts to salvage P values by detaching them from their use in significance tests. One approach is to focus on P values as continuous measures of compatibility, as described earlier. Although this approach has its own limitations (as described in points 1, 2, 5, 9, 15, 18, 19), it avoids comparison of P values with arbitrary cutoffs such as 0.05, (as described in 3, 4, 6–8, 10–13, 15, 16, 21 and 23–25). Another approach is to teach and use correct relations of P values to hypothesis probabilities. For example, under common statistical models, one-sided P values can provide lower bounds on probabilities for hypotheses about effect directions [ 45 , 46 , 112 , 113 ]. Whether such reinterpretations can eventually replace common misinterpretations to good effect remains to be seen.

A shift in emphasis from hypothesis testing to estimation has been promoted as a simple and relatively safe way to improve practice [ 5 , 61 , 63 , 114 , 115 ] resulting in increasing use of confidence intervals and editorial demands for them; nonetheless, this shift has brought to the fore misinterpretations of intervals such as 19–23 above [ 116 ]. Other approaches combine tests of the null with further calculations involving both null and alternative hypotheses [ 117 , 118 ]; such calculations may, however, may bring with them further misinterpretations similar to those described above for power, as well as greater complexity.

Meanwhile, in the hopes of minimizing harms of current practice, we can offer several guidelines for users and readers of statistics, and re-emphasize some key warnings from our list of misinterpretations:

• Correct and careful interpretation of statistical tests demands examining the sizes of effect estimates and confidence limits, as well as precise P values (not just whether P values are above or below 0.05 or some other threshold).
• Careful interpretation also demands critical examination of the assumptions and conventions used for the statistical analysis—not just the usual statistical assumptions, but also the hidden assumptions about how results were generated and chosen for presentation.
• It is simply false to claim that statistically nonsignificant results support a test hypothesis, because the same results may be even more compatible with alternative hypotheses—even if the power of the test is high for those alternatives.
• Interval estimates aid in evaluating whether the data are capable of discriminating among various hypotheses about effect sizes, or whether statistical results have been misrepresented as supporting one hypothesis when those results are better explained by other hypotheses (see points 4–6). We caution however that confidence intervals are often only a first step in these tasks. To compare hypotheses in light of the data and the statistical model it may be necessary to calculate the P value (or relative likelihood) of each hypothesis. We further caution that confidence intervals provide only a best-case measure of the uncertainty or ambiguity left by the data, insofar as they depend on an uncertain statistical model.
• Correct statistical evaluation of multiple studies requires a pooled analysis or meta-analysis that deals correctly with study biases [ 68 , 119 – 125 ]. Even when this is done, however, all the earlier cautions apply. Furthermore, the outcome of any statistical procedure is but one of many considerations that must be evaluated when examining the totality of evidence. In particular, statistical significance is neither necessary nor sufficient for determining the scientific or practical significance of a set of observations. This view was affirmed unanimously by the U.S. Supreme Court, (Matrixx Initiatives, Inc., et al. v. Siracusano et al. No. 09–1156. Argued January 10, 2011, Decided March 22, 2011), and can be seen in our earlier quotes from Neyman and Pearson.
• Any opinion offered about the probability , likelihood , certainty , or similar property for a hypothesis cannot be derived from statistical methods alone. In particular, significance tests and confidence intervals do not by themselves provide a logically sound basis for concluding an effect is present or absent with certainty or a given probability. This point should be borne in mind whenever one sees a conclusion framed as a statement of probability, likelihood, or certainty about a hypothesis. Information about the hypothesis beyond that contained in the analyzed data and in conventional statistical models (which give only data probabilities) must be used to reach such a conclusion; that information should be explicitly acknowledged and described by those offering the conclusion. Bayesian statistics offers methods that attempt to incorporate the needed information directly into the statistical model; they have not, however, achieved the popularity of P values and confidence intervals, in part because of philosophical objections and in part because no conventions have become established for their use.
• All statistical methods (whether frequentist or Bayesian, or for testing or estimation, or for inference or decision) make extensive assumptions about the sequence of events that led to the results presented—not only in the data generation, but in the analysis choices. Thus, to allow critical evaluation, research reports (including meta-analyses) should describe in detail the full sequence of events that led to the statistics presented, including the motivation for the study, its design, the original analysis plan, the criteria used to include and exclude subjects (or studies) and data, and a thorough description of all the analyses that were conducted.

In closing, we note that no statistical method is immune to misinterpretation and misuse, but prudent users of statistics will avoid approaches especially prone to serious abuse. In this regard, we join others in singling out the degradation of P values into “significant” and “nonsignificant” as an especially pernicious statistical practice [ 126 ].

Acknowledgments

SJS receives funding from the IDEAL project supported by the European Union’s Seventh Framework Programme for research, technological development and demonstration under Grant Agreement No. 602552. We thank Stuart Hurlbert, Deborah Mayo, Keith O’Rourke, and Andreas Stang for helpful comments, and Ron Wasserstein for his invaluable encouragement on this project.

Editor’s note

This article has been published online as supplementary material with an article of Wasserstein RL, Lazar NA. The ASA’s statement on p-values: context, process and purpose. The American Statistician 2016.

Albert Hofman, Editor-in-Chief EJE.

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What Is P-Value?

Understanding p-value.

• P-Value in Hypothesis Testing

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P-Value: What It Is, How to Calculate It, and Why It Matters

Yarilet Perez is an experienced multimedia journalist and fact-checker with a Master of Science in Journalism. She has worked in multiple cities covering breaking news, politics, education, and more. Her expertise is in personal finance and investing, and real estate.

In statistics, a p-value indicates the likelihood of obtaining a value equal to or greater than the observed result if the null hypothesis is true.

The p-value serves as an alternative to rejection points to provide the smallest level of significance at which the null hypothesis would be rejected. A smaller p-value means stronger evidence in favor of the alternative hypothesis.

P-value is often used to promote credibility for studies or reports by government agencies. For example, the U.S. Census Bureau stipulates that any analysis with a p-value greater than 0.10 must be accompanied by a statement that the difference is not statistically different from zero. The Census Bureau also has standards in place stipulating which p-values are acceptable for various publications.

Key Takeaways

• A p-value is a statistical measurement used to validate a hypothesis against observed data.
• A p-value measures the probability of obtaining the observed results, assuming that the null hypothesis is true.
• The lower the p-value, the greater the statistical significance of the observed difference.
• A p-value of 0.05 or lower is generally considered statistically significant.
• P-value can serve as an alternative to—or in addition to—preselected confidence levels for hypothesis testing.

Jessica Olah / Investopedia

P-values are usually calculated using statistical software or p-value tables based on the assumed or known probability distribution of the specific statistic tested. While the sample size influences the reliability of the observed data, the p-value approach to hypothesis testing specifically involves calculating the p-value based on the deviation between the observed value and a chosen reference value, given the probability distribution of the statistic. A greater difference between the two values corresponds to a lower p-value.

Mathematically, the p-value is calculated using integral calculus from the area under the probability distribution curve for all values of statistics that are at least as far from the reference value as the observed value is, relative to the total area under the probability distribution curve. Standard deviations, which quantify the dispersion of data points from the mean, are instrumental in this calculation.

The calculation for a p-value varies based on the type of test performed. The three test types describe the location on the probability distribution curve: lower-tailed test, upper-tailed test, or two-tailed test . In each case, the degrees of freedom play a crucial role in determining the shape of the distribution and thus, the calculation of the p-value.

In a nutshell, the greater the difference between two observed values, the less likely it is that the difference is due to simple random chance, and this is reflected by a lower p-value.

The P-Value Approach to Hypothesis Testing

The p-value approach to hypothesis testing uses the calculated probability to determine whether there is evidence to reject the null hypothesis. This determination relies heavily on the test statistic, which summarizes the information from the sample relevant to the hypothesis being tested. The null hypothesis, also known as the conjecture, is the initial claim about a population (or data-generating process). The alternative hypothesis states whether the population parameter differs from the value of the population parameter stated in the conjecture.

In practice, the significance level is stated in advance to determine how small the p-value must be to reject the null hypothesis. Because different researchers use different levels of significance when examining a question, a reader may sometimes have difficulty comparing results from two different tests. P-values provide a solution to this problem.

Even a low p-value is not necessarily proof of statistical significance, since there is still a possibility that the observed data are the result of chance. Only repeated experiments or studies can confirm if a relationship is statistically significant.

For example, suppose a study comparing returns from two particular assets was undertaken by different researchers who used the same data but different significance levels. The researchers might come to opposite conclusions regarding whether the assets differ.

If one researcher used a confidence level of 90% and the other required a confidence level of 95% to reject the null hypothesis, and if the p-value of the observed difference between the two returns was 0.08 (corresponding to a confidence level of 92%), then the first researcher would find that the two assets have a difference that is statistically significant , while the second would find no statistically significant difference between the returns.

To avoid this problem, the researchers could report the p-value of the hypothesis test and allow readers to interpret the statistical significance themselves. This is called a p-value approach to hypothesis testing. Independent observers could note the p-value and decide for themselves whether that represents a statistically significant difference or not.

Example of P-Value

An investor claims that their investment portfolio’s performance is equivalent to that of the Standard & Poor’s (S&P) 500 Index . To determine this, the investor conducts a two-tailed test.

The null hypothesis states that the portfolio’s returns are equivalent to the S&P 500’s returns over a specified period, while the alternative hypothesis states that the portfolio’s returns and the S&P 500’s returns are not equivalent—if the investor conducted a one-tailed test , the alternative hypothesis would state that the portfolio’s returns are either less than or greater than the S&P 500’s returns.

The p-value hypothesis test does not necessarily make use of a preselected confidence level at which the investor should reset the null hypothesis that the returns are equivalent. Instead, it provides a measure of how much evidence there is to reject the null hypothesis. The smaller the p-value, the greater the evidence against the null hypothesis.

Thus, if the investor finds that the p-value is 0.001, there is strong evidence against the null hypothesis, and the investor can confidently conclude that the portfolio’s returns and the S&P 500’s returns are not equivalent.

Although this does not provide an exact threshold as to when the investor should accept or reject the null hypothesis, it does have another very practical advantage. P-value hypothesis testing offers a direct way to compare the relative confidence that the investor can have when choosing among multiple different types of investments or portfolios relative to a benchmark such as the S&P 500.

For example, for two portfolios, A and B, whose performance differs from the S&P 500 with p-values of 0.10 and 0.01, respectively, the investor can be much more confident that portfolio B, with a lower p-value, will actually show consistently different results.

Is a 0.05 P-Value Significant?

A p-value less than 0.05 is typically considered to be statistically significant, in which case the null hypothesis should be rejected. A p-value greater than 0.05 means that deviation from the null hypothesis is not statistically significant, and the null hypothesis is not rejected.

What Does a P-Value of 0.001 Mean?

A p-value of 0.001 indicates that if the null hypothesis tested were indeed true, then there would be a one-in-1,000 chance of observing results at least as extreme. This leads the observer to reject the null hypothesis because either a highly rare data result has been observed or the null hypothesis is incorrect.

How Can You Use P-Value to Compare 2 Different Results of a Hypothesis Test?

If you have two different results, one with a p-value of 0.04 and one with a p-value of 0.06, the result with a p-value of 0.04 will be considered more statistically significant than the p-value of 0.06. Beyond this simplified example, you could compare a 0.04 p-value to a 0.001 p-value. Both are statistically significant, but the 0.001 example provides an even stronger case against the null hypothesis than the 0.04.

The p-value is used to measure the significance of observational data. When researchers identify an apparent relationship between two variables, there is always a possibility that this correlation might be a coincidence. A p-value calculation helps determine if the observed relationship could arise as a result of chance.

U.S. Census Bureau. “ Statistical Quality Standard E1: Analyzing Data .”

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