null hypothesis critical value calculator

Critical Value Calculator

Use this calculator for critical values to easily convert a significance level to its corresponding Z value, T score, F-score, or Chi-square value. Outputs the critical region as well. The tool supports one-tailed and two-tailed significance tests / probability values.

Related calculators

• Using the critical value calculator
• What is a critical value?
• T critical value calculation
• Z critical value calculation
• F critical value calculation

    Using the critical value calculator

If you want to perform a statistical test of significance (a.k.a. significance test, statistical significance test), determining the value of the test statistic corresponding to the desired significance level is necessary. You need to know the desired error probability ( p-value threshold , common values are 0.05, 0.01, 0.001) corresponding to the significance level of the test. If you know the significance level in percentages, simply subtract it from 100%. For example, 95% significance results in a probability of 100%-95% = 5% = 0.05 .

Then you need to know the shape of the error distribution of the statistic of interest (not to be mistaken with the distribution of the underlying data!) . Our critical value calculator supports statistics which are either:

• Z -distributed (normally distributed, e.g. absolute difference of means)
• T -distributed (Student's T distribution, usually appropriate for small sample sizes, equivalent to the normal for sample sizes over 30)
• X 2 -distributed ( Chi square distribution, often used in goodness-of-fit tests, but also for tests of homogeneity or independence)
• F -distributed (Fisher-Snedecor distribution), usually used in analysis of variance (ANOVA)

Then, for distributions other than the normal one (Z), you need to know the degrees of freedom . For the F statistic there are two separate degrees of freedom - one for the numerator and one for the denominator.

Finally, to determine a critical region, one needs to know whether they are testing a point null versus a composite alternative (on both sides) or a composite null versus (covering one side of the distribution) a composite alternative (covering the other). Basically, it comes down to whether the inference is going to contain claims regarding the direction of the effect or not. Should one want to claim anything about the direction of the effect, the corresponding null hypothesis is direction as well (one-sided hypothesis).

Depending on the type of test - one-tailed or two-tailed, the calculator will output the critical value or values and the corresponding critical region. For one-sided tests it will output both possible regions, whereas for a two-sided test it will output the union of the two critical regions on the opposite sides of the distribution.

    What is a critical value?

A critical value (or values) is a point on the support of an error distribution which bounds a critical region from above or below. If the statistics falls below or above a critical value (depending on the type of hypothesis, but it has to fall inside the critical region) then a test is declared statistically significant at the corresponding significance level. For example, in a two-tailed Z test with critical values -1.96 and 1.96 (corresponding to 0.05 significance level) the critical regions are from -∞ to -1.96 and from 1.96 to +∞. Therefore, if the statistic falls below -1.96 or above 1.96, the null hypothesis test is statistically significant.

You can think of the critical value as a cutoff point beyond which events are considered rare enough to count as evidence against the specified null hypothesis. It is a value achieved by a distance function with probability equal to or greater than the significance level under the specified null hypothesis. In an error-probabilistic framework, a proper distance function based on a test statistic takes the generic form [1] :

X (read "X bar") is the arithmetic mean of the population baseline or the control, μ 0 is the observed mean / treatment group mean, while σ x is the standard error of the mean (SEM, or standard deviation of the error of the mean).

Here is how it looks in practice when the error is normally distributed (Z distribution) with a one-tailed null and alternative hypotheses and a significance level α set to 0.05:

And here is the same significance level when applied to a point null and a two-tailed alternative hypothesis:

The distance function would vary depending on the distribution of the error: Z, T, F, or Chi-square (X 2 ). The calculation of a particular critical value based on a supplied probability and error distribution is simply a matter of calculating the inverse cumulative probability density function (inverse CPDF) of the respective distribution. This can be a difficult task, most notably for the T distribution [2] .

    T critical value calculation

The T-distribution is often preferred in the social sciences, psychiatry, economics, and other sciences where low sample sizes are a common occurrence. Certain clinical studies also fall under this umbrella. This stems from the fact that for sample sizes over 30 it is practically equivalent to the normal distribution which is easier to work with. It was proposed by William Gosset, a.k.a. Student, in 1908 [3] , which is why it is also referred to as "Student's T distribution".

To find the critical t value, one needs to compute the inverse cumulative PDF of the T distribution. To do that, the significance level and the degrees of freedom need to be known. The degrees of freedom represent the number of values in the final calculation of a statistic that are free to vary whilst the statistic remains fixed at a certain value.

It should be noted that there is not, in fact, a single T-distribution, but there are infinitely many T-distributions, each with a different level of degrees of freedom. Below are some key values of the T-distribution with 1 degree of freedom, assuming a one-tailed T test is to be performed. These are often used as critical values to define rejection regions in hypothesis testing.

    Z critical value calculation

The Z-score is a statistic showing how many standard deviations away from the normal, usually the mean, a given observation is. It is often called just a standard score, z-value, normal score, and standardized variable. A Z critical value is just a particular cutoff in the error distribution of a normally-distributed statistic.

Z critical values are computed by using the inverse cumulative probability density function of the standard normal distribution with a mean (μ) of zero and standard deviation (σ) of one. Below are some commonly encountered probability values (significance levels) and their corresponding Z values for the critical region, assuming a one-tailed hypothesis .

The critical region defined by each of these would span from the Z value to plus infinity for the right-tailed case, and from minus infinity to minus the Z critical value in the left-tailed case. Our calculator for critical value will both find the critical z value(s) and output the corresponding critical regions for you.

Chi Square (Χ 2 ) critical value calculation

Chi square distributed errors are commonly encountered in goodness-of-fit tests and homogeneity tests, but also in tests for independence in contingency tables. Since the distribution is based on the squares of scores, it only contains positive values. Calculating the inverse cumulative PDF of the distribution is required in order to convert a desired probability (significance) to a chi square critical value.

Just like the T and F distributions, there is a different chi square distribution corresponding to different degrees of freedom. Hence, to calculate a Χ 2 critical value one needs to supply the degrees of freedom for the statistic of interest.

    F critical value calculation

F distributed errors are commonly encountered in analysis of variance (ANOVA), which is very common in the social sciences. The distribution, also referred to as the Fisher-Snedecor distribution, only contains positive values, similar to the Χ 2 one. Similar to the T distribution, there is no single F-distribution to speak of. A different F distribution is defined for each pair of degrees of freedom - one for the numerator and one for the denominator.

Calculating the inverse cumulative PDF of the F distribution specified by the two degrees of freedom is required in order to convert a desired probability (significance) to a critical value. There is no simple solution to find a critical value of f and while there are tables, using a calculator is the preferred approach nowadays.

    References

1 Mayo D.G., Spanos A. (2010) – "Error Statistics", in P. S. Bandyopadhyay & M. R. Forster (Eds.), Philosophy of Statistics, (7, 152–198). Handbook of the Philosophy of Science . The Netherlands: Elsevier.

2 Shaw T.W. (2006) – "Sampling Student's T distribution – use of the inverse cumulative distribution function", Journal of Computational Finance 9(4):37-73, DOI:10.21314/JCF.2006.150

3 "Student" [William Sealy Gosset] (1908) - "The probable error of a mean", Biometrika 6(1):1–25. DOI:10.1093/biomet/6.1.1

Cite this calculator & page

If you'd like to cite this online calculator resource and information as provided on the page, you can use the following citation: Georgiev G.Z., "Critical Value Calculator" , [online] Available at: https://www.gigacalculator.com/calculators/critical-value-calculator.php URL [Accessed Date: 07 Sep, 2024].

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The author of this tool

     Statistical calculators

Hypothesis Testing Calculator

Type II Error

The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.

Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.

To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.

In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.

To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.

When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.

Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.

Critical Value Calculator

To get the result, fill out the calculator form and press the Calculate button.

Table of Content

• 1 What is Critical Value
• 2 Critical Value Formula
• 3 How to calculate critical value? - steps and process
• 4 Common confidence levels and their critical values
• 5 Types of Critical Values
• 6 Critical Value of Z
• 7 Assistance offered by this critical value calculator

What is Critical Value

Any point on a line that divides the graph into two equal parts is considered to have a critical value, to put it simply. Depending on the area in which the value falls, the null hypothesis is either rejected or accepted. One of the two divisions created by the critical value is known as the rejection region. The null hypothesis would not be accepted if the test value were to be found in the rejection region.

Critical Value Formula

The critical value is the point on a test statistic distribution where it is decided whether to reject or not to reject the null hypothesis. The critical value formula depends on the kind of test being run and the level of significance (alpha) picked.

For Example, the critical value formula for a two-tailed z-test with a normal distribution is:

where "alpha" denotes the level of significance and "z" denotes the standard normal distribution (usually 0.05).

The formula for the critical value in a two-tailed t-test is:

where "df" stands for degrees of freedom, "alpha" for level of significance, and "t" stands for the t-distribution.

The formula for a one-tailed test varies depending on which side of the distribution you want to test.

It's crucial to remember that while these formulas provide the critical value for a specific level of significance, the critical value for a specific probability level can also be determined by using the inverse of the cumulative distribution function of the test statistic.

How to Calculate Critical Values: A Step-by-Step Guide

Calculating critical values is an important part of statistical analysis, and understanding the process can help you make better decisions based on data. In this guide, we'll walk you through the steps for calculating critical values.

Step 1: Determine the Type of Hypothesis Test

The first step is to determine whether you are conducting a one-tailed or two-tailed hypothesis test. In a one-tailed test, the null hypothesis is that there is no effect or a specific direction of effect (i.e., "greater than" or "less than"). In a two-tailed test, the null hypothesis is that there is no effect, without specifying the direction of the effect.

Step 2: Choose the Level of Significance

The level of significance, denoted by α (alpha), is the probability of rejecting the null hypothesis when it is true. Common levels of significance are 0.05 (5%) and 0.01 (1%), but the specific value depends on the researcher's preference and the context of the study.

Step 3: Determine the Degrees of Freedom

The degrees of freedom, denoted by df, represent the number of independent pieces of information in the sample that can vary. The formula for degrees of freedom depends on the type of test and the sample size.

For a one-tailed test with a sample size of n, df = n - 1.

For example , if you have a sample size of n = 20, the degrees of freedom for a one-tailed test would be df = 20 - 1 = 19.

For a two-tailed test with a sample size of n, df = n - 2.

For example , if you have a sample size of n = 30, the degrees of freedom for a two-tailed test would be df = 30 - 2 = 28.

Step 4: Look Up the Critical Value

Once you know the type of test, level of significance, and degrees of freedom, you can find the critical value from a statistical table. The critical value is the minimum value of the test statistic that will lead to the rejection of the null hypothesis.

For example , suppose you are conducting a one-tailed test with a level of significance α = 0.05 and degrees of freedom df = 19. From a t-distribution table, the critical value is 1.734.

For a two-tailed test, you need to find the critical value for both tails. The critical values are typically denoted by tα/2 and -tα/2

For example , if α = 0.01 and df = 28, the critical values from a t-distribution table are -2.763 and 2.763, respectively.

Step 5: Calculate the Test Statistic

Calculate the test statistic using the sample data and the null hypothesis. The test statistic is the value used to determine whether to reject or fail to reject the null hypothesis.

For example , suppose you are testing the null hypothesis that the mean weight of a certain population is 50 kg, and your sample mean is 55 kg with a sample standard deviation of 10 kg. The test statistic for a one-tailed test is calculated as:

t = (sample mean - null hypothesis) / (sample standard deviation/sqrt (sample size)) = (55 - 50) / (10 / sqrt(20)) = 3.162

Since the test statistic of 3.162 is greater than the critical value of 1.734, we reject the null hypothesis at the 0.05 level of significance. This means that we have evidence to support the alternative hypothesis that the mean weight of the population is greater than 50 kg.

Common confidence levels and their critical values

In statistical inference, confidence levels are used to quantify the uncertainty associated with estimating a population parameter based on a sample. A confidence level is the probability that a statistical interval, such as a confidence interval, contains the true population parameter. Common confidence levels used in practice include 90%, 95%, and 99%.

The critical values for a given confidence level depend on the distribution of the test statistic and the degrees of freedom. Let's overview some examples of common confidence levels and their corresponding critical values for different distributions.

Normal Distribution

For a normal distribution, the critical values for different confidence levels are given by the z-score. The z-score is the number of standard deviations a value is away from the mean. The critical values for different confidence levels are:

90% confidence level: z = 1.645

95% confidence level: z = 1.96

99% confidence level: z = 2.576

Example : If you want to construct a 95% confidence interval for the population mean based on a sample from a normal distribution, you would use the critical value of z = 1.96

Student's t-Distribution

For small sample sizes or when the population standard deviation is unknown, the t-distribution is used instead of the normal distribution. The critical values for different confidence levels for the t-distribution depend on the degrees of freedom. Some common values are:

90% confidence level: t(df) = 1.645

95% confidence level: t(df) = 1.96

99% confidence level: t(df) = 2.576

Example : If you want to construct a 90% confidence interval for the population mean based on a sample from a t-distribution with 10 degrees of freedom, you would use the critical value of t(10) = 1.645

Chi-Squared Distribution

The chi-squared distribution is used for hypothesis tests and confidence intervals involving the variance of a normally distributed population. The critical values for different confidence levels depend on the degrees of freedom. Some common values are:

90% confidence level: χ²(df) = 14.68

95% confidence level: χ²(df) = 16.92

99% confidence level: χ²(df) = 23.59

Example : If you want to construct a 99% confidence interval for the population variance based on a sample from a normal distribution, you would use the critical value of χ2(n-1) = 23.59, where n is the sample size.

Types of Critical Values

Critical values are specific values that are used to determine whether to reject or fail to reject the null hypothesis. There are different types of critical values used in different statistical tests, and understanding these types can help in making accurate statistical inferences. Below are some types of critical values:

One-tailed critical values

One-tailed critical values are used in hypothesis testing when the alternative hypothesis is directional. To put it another way, the test is made to see if the sample mean differs significantly from the population mean.

Two-tailed critical values

Two-tailed critical values are used in hypothesis testing when the alternative hypothesis is non-directional. Two-tailed critical values are located in the middle of the distribution and correspond to a specified level of significance split between the two tails.

Upper-tailed critical values

Upper-tailed critical values are used in hypothesis testing when the test is designed to determine if the sample mean is significantly greater than the population mean. Upper-tailed critical values are located at the extreme right end of the distribution and correspond to a specified level of significance.

Lower-tailed critical values

Lower-tailed critical values are used in hypothesis testing when the test is designed to determine if the sample mean is significantly less than the population means. A certain level of significance is corresponding to lower-tailed critical values, which are situated at the extreme left end of the distribution.

Critical Value of Z

The critical value of z is a term used in statistics to indicate the value of the standard normal distribution that corresponds to a particular level of significance or alpha (α). The standard normal distribution is a continuous probability distribution that is often used to model random variables that are approximately normal.

As part of a statistical procedure called hypothesis testing, which determines whether there is enough data to support or disprove a null hypothesis, the critical value of z is frequently used. The null hypothesis is a statement that assumes that there is no significant difference between two or more populations or sets of data. For example, suppose you want to perform a two-tailed hypothesis test with a level of significance of 0.05 (i.e., α = 0.05).

The critical value of z at this level of significance is ±1.96. It means that if the test statistic falls outside of the range of -1.96 to 1.96, we can reject the null hypothesis and conclude that there is sufficient evidence to support the alternative hypothesis.

Assistance offered by this critical value calculator

• Calculating critical values for various statistical distributions
• Customizable inputs
• User-friendly interface
• Results in multiple formats

With customizable inputs, users can specify the significance level, degrees of freedom, and other parameters necessary to obtain accurate results. The user-friendly interface of our calculator makes it easy to enter inputs and view results.

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Critical values calculator :

Welcome to our critical value calculator. This tool is used to find the left, right and two-tailed critical value for z, t-student, chi-square, r and F-distribution. Whether you're a student, researcher, or professional, this tool is designed to meet your statistical needs.

The critical value is the most important component in hypothesis testing as well as in the confidence interval. If the hypothesis test is one-tailed then it gives one critical value and if the hypothesis test is two-tailed then it gives two critical values i.e. one is positive and the other is negative.

Table of Contents

• How to Use the Calculator
• Related Calculators

What are Critical Values?

• Different Types of Critical Values

How to find value from table ?

Where does critical values are used , practical examples:, how to use the critical values calculator.

The critical value calculator is an easy-to-use tool for determining critical value whether it is a one-tailed test or two-tailed test.

• Test type : in this fields enter the type of test from z-score, t-score, chi-square \( \chi^2 \), F-score or correlation coefficient r.
• For T Critical Value: Enter the degrees of freedom and the significance level \(\alpha\).
• For Z Critical Value: Enter the significance level \(\alpha\).
• For Chi-Square Critical Value: Enter the degrees of freedom and the significance level \(\alpha\).
• For F Critical Value: Enter the degrees of freedom for both the numerator and the denominator, and the significance level \(\alpha\).
• For r Critical Value: Enter sample size, and the significance level \(\alpha\).
• Calculate : Click on calculate button to get the desired critical value.

Related Calculators :

Below are more calculators which use the critical value to perform statistical analysis.

The critical value also known as critical point is used to decide whether to reject or fail to reject the null hypothesis while hypothesis testing. It is also used to find the lower and upper limits of the confidence interval. The critical value has left-tailed, right-tailed, and two-tailed critical values.

• Left-Tailed Test : As this is a left-tailed test, if the test statistic falls to the left of the critical value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
• Right-Tailed Test : As this is a right-tailed test, if the test statistic falls to the right of the critical value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
• Two-Tailed Test : As this is a two-tailed test, if the test statistic falls in the left or right tail beyond the critical value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.

The above figure shows left, right and two tailed critical value along with there rejection region for z distribution.

Different types of critical values

The different types of critical values are explained below.

t critical value :

In t distribution, the significance level \( ( \alpha ) \) and degrees of freedom are required to find the critical values. This distribution is used when the population standard deviation is unknown.

Z Critical Value :

In z distribution, the only significance level is required to find the critical values. This distribution is used when the population standard deviation is known.

Chi-Square Critical Value :

F critical value :.

In f test the significance level \( ( \alpha ) \) and the degrees of freedom for the denominator and numerator are used to find the critical value. Note in the f test there are two degrees of freedom i.e \( df_1 \) and \( df_2 \)

Correlation Coefficient (r) critical Value

The sample size and significance level \( \alpha \) are used to find the critical value. The critical values for the correlation coefficient are used to determine the significance of the correlation between two variables.

There are numbers in critical value tables, but the procedure is the same to find the critical value. Let's take one example of how to find the critical value on the z table.

Step 1 : Determine the Significance Level \( (\alpha) \)

• If the hypothesis test is a two tailed test, divide \( \alpha \) by 2.
• If the hypothesis test is a one tailed test, use \( \alpha \) directly.

Step 2 : Calculate the Critical Probability:

• For a left-tail test: Use the significance level.
• For a right-tail test: Subtract the significance level from 1.
• For two-tailed tests: Use both the above probabilities.

Step 3 : Locate the Critical Value:

• Use a Z table to find the Z-score corresponding to the calculated probability.
• The Z table provides the area to the left of a Z-score in a standard normal distribution.

Critical value table :

Left tailed critical value table for z test

Right tailed critical value table for z test

Two tailed critical value table for z test

The critical values are used in various fields like statistical hypothesis testing and research. Here’s a summary of where they are applied:

Decision Making:

Based on the calculated test statistic ( T, Z, r, Chi-Square), critical values determine whether to reject or fail to reject the null hypothesis.

Significance Levels :

Critical values, which are usually set at \(0.05\) or \(0.01\), aid in assessing the degree of confidence in the statistical findings.

Two-Tailed vs. One-Tailed Tests:

Depending on whether the test is two-tailed (non-directional hypothesis) or one-tailed (directional hypothesis), different critical values apply.

The critical values are used in medical research for testing the effectiveness of a new drug compared to a placebo.

It can be used in market research for analyzing survey data to determine if there is a preference between two products.

It can also be used in quality control to assess if a manufacturing process meets specified standards.

Why Use Our Critical Values Calculator ?

• Accuracy: Our calculator provides precise critical values for your statistical tests.
• User-Friendly: The interface is intuitive, making it easy to input data and obtain results quickly.
• Versatile : Calculate T, Z, Chi-Square,r, and F critical values in one place.
• Free and Accessible: No cost and accessible online anytime, anywhere.

• chi-square value

How does t critical value calculator work?

• Enter Significance Level(α) In The Input Box.
• Put the  Degrees Of Freedom In The Input Box.
• Hit The Calculate Button To Find t Critical Value.
• Use The Reset Button To calculate New Values.

How does z critical value calculator work?

• Enter The Significance Level(α) In The Input Box.
• Use The Calculate Button To Get The Z Critical Value.

How does R critical value calculator work?

• Enter Significance Level(α) & Degree of freedom In The Input Boxes.
• Click The Calculate Button.
• Hit The Reset Button To calculate New Values.

How does this calculator work?

• Enter Significance Level(α) & Degree of freedom In Required Input Boxes.
• Press The Reset Button To calculate New Values.

How does F critical value calculator work?

• Enter Significance Level(α)
• Enter Degree of freedom of numerator and denominator in required input boxes.

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Other Calculators

Critical value calculator.

Critical t value calculator enables you to calculate the critical value of z and t at one click. You don’t have to look into hundreds of values in t table or a z table because this z critical value calculator calculates critical values in real time . Keep on reading if you are interested in critical value definition, the difference between t and z critical value, and how to calculate the critical value of t and z without using a critical values calculator.

 Table of Content

What is a critical value, critical value formula, how to find critical values.

• T-Distribution Table

A  critical value is a point on the t-distribution that is compared to the test statistic to determine whether to reject the null hypothesis in hypothesis testing. If the absolute value of the test statistic is greater than the critical value, statistical significance can be declared as well as the null hypothesis can be rejected. Critical value tests can be:

Left-tailed test: Q (α)

Right-tailed test: Q (1 - α)

Two-tailed test: Q (α/2)] ∪ Q (1 - α/2)

What is a t critical value?

T critical value is a point that cuts off the student t distribution . T value is used in a hypothesis test to compare against a calculated t score. The critical value of t helps to decide if a null hypothesis should be supported or rejected.

What is a z critical value?

Z critical value is a point that cuts off an area under the standard normal distribution . The critical value of z can tell what probability any particular variable will have. Z and t critical values are almost identical.

What is f critical value?

F critical value is a value at which the threshold probability α of type-I error (reject a true null hypothesis mistakenly).  The f statistics is the value that follows the f-distribution table.

Here are a few tests that help to calculate the f values.

• Overall significance in regression analysis. k
• Compare two nested regression models. 
• The equality of variances in two normally distributed populations.

All the above tests are right-tailed. F critical value calculator above will help you to calculate the f critical value with a single click.

What is the chi-square value?

In certain hypothesis tests and confidence intervals, chi-square values are thresholds for statistical significance. The Chi-square distribution table is used to evaluate the chi-square critical values. It is rather tough to calculate the critical value by hand, so try a reference table or chi-square critical value calculator above.

The chi-square critical values are always positive and can be used in the following tests.

• Goodness-of-fit tests
• Homogeneity tests
• Tests for independence in contingency tables

Unlike the t & f critical value, Χ 2  (chi-square) critical value needs to supply the degrees of freedom to get the result.

The formula of z and t critical value can be expressed as:

• Q t is the quantile function of t student distribution
• u is the quantile function of the normal distribution
• d refers to the degree of freedom
• α is the significance level

A critical value of t calculator uses all these formulas to produce the exact critical values needed to accept or reject a hypothesis.

Calculating critical value is a tiring task because it involves looking for values into the t-distribution chart. The t-distribution table (student t-test distribution) consists of hundreds of values, so, it is convenient to use t table value calculator above for critical values. However, if you want to find critical values without using t table calculator, follow the examples given below.

How to find t critical value?

Find the t critical value if the size of the sample is 5 and the significance level is 0.05 .

Subtract 1 from the sample size to get the degree of freedom. Degree of Freedom = N – 1 = 5 – 1 Degree of freedom = 4 α = 0.05

Depending on the test, choose the one-tailed t distribution table or two-tailed t table below.

Look for the degree of freedom in the most left column. Also, look for the significance level α in the top row. Pick the value occurring at the intersection of the mentioned row and column. In this case, the t critical value is 2.132 .

How to find z critical value?

Find the z critical value if the significance level is 0.02 .

Divide the significance level α by 2 α/2 = 0.02/2 α/2 = 0.01

Subtract α/2 from 1. 1 - α/2 = 1 – 0.01 1 - α/2 = 0.99

Search the value 0.99 in the z table given below. Add the values of intersecting row (top) and column (most left) to get the z critical value. 2.3 + 0.03 = 2.33 Z critical value = ±2.33 for the two-tailed test.

T-Distribution Table (One Tail)

The t table for one-tail probability is given below.

T-Distribution Table (Two Tail)

The t table for two-tail probability is given below.

Z table (right-tailed)

The normal distribution table for the right-tailed test is given below.

Z table (left-tailed)

The normal distribution table for the left-tailed test is given below.

• Krista King Math | Online math tutor. 2021. Critical points and the first derivative test — Krista King Math | Online math tutor.
• S.3.1 Hypothesis Testing (Critical Value Approach) | STAT ONLINE. PennState: Statistics Online Courses.
• 7.1.3.1. Critical values and p values. National Institute of Standards and Technology (NIST).

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Critical Value Calculator

Enter the significant level along with degrees of freedom and the tool will try to figure out critical values for T, Z, Chi, and F distributions.

Chi-Square Value

How Does T Critical Value Calculator Work?

• Enter Significance Level(α) In The Input Box.
• Put the  Degrees Of Freedom In The Input Box.
• Enter Degree of freedom denominator in required input box.
• Hit The Calculate Button To Find T Critical Value.

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Get the critical values associated with a particular significance level (alpha) and statistical distributions with the critical value calculator.

The calculator functions to provide left-tailed, right-tailed, and two-tailed P values for Chi-square, T, F, R, and Z score values that help you better judge the critical points of the given probability density function.

What Is a Critical Value?

A critical value is said to be as a line on a graph that divides a distribution graph into sections that indicate ‘rejection regions.’ Generally, if a test value falls into a rejection rejoin, then it means that an accepted hypothesis (represent as a null hypothesis) should be rejected.

Left-tailed: Q (α)

Right-tailed: Q (1 - α)

Two-tailed: Q (α/2)] ∪ Q (1 - α/2)

How to Calculate Critical Value With This Tool

Simply, you just have to follow the given steps:

Find Critical Value for T

• At first, you ought to select the option “Critical value for t” from the drop-down list
• Now, you just have to add the value of the “significance level” into the designated field
• Finally, you have to add the value of “degrees of freedom” into the designated field

Outputs: After adding values into the above fields, just hit the calculate button:

• Critical value for t (Right Tailed)
• Critical value for t (Two Tailed)

Find Critical Value For Z

• You just have to choose the option “Critical value for z” form the drop-down menu of this tool
• Right after, you ought to add the value of the “significance level” into the given box

Outputs: Now, hit the calculate button, this z value calculator will show:

• Critical value for z (Right Tailed)
• Critical value for z (Two Tailed)

Find Critical Value for Chi-Square

• First, choose the option “Critical value for chi-square” form the list of drop-down
• Then, simply add the value for a “significance level” into the above-designated box
• Very next, add the value for a “degrees of freedom” into the given field of calculator

Outputs: Now, you have to make a click on the calculate button to calculate chi square value for the distribution, the tool generates:

• Chi-Square critical value (Right Tailed)
• Chi-Square critical value (Two Tailed)
• P value (for the chi square distribution)

Find Critical Value For F

• First, choose the option of “Critical value for f” from the given drop-down menu
• Very next, you have to enter the value of a “degrees of freedom 1” into the designated field
• Right after, you ought to add the value of a “degrees if freedom 2” into the given box
• Finally, put the value of “significance level” into the designated box

Outputs: Once done, click on the calculate button, this f value calculator will generate:

• Critical value f (Right Tailed)
• Critical value f (Two Tailed)

Z Score Table (Right):

The z-table is the normal distribution shows the area to the right-hand side of the curve. You can use these values to determine the area between z=0 and any positive (+) value.

Z Score Table (Left):

The left z-table shows the area to the left of Z.

T Critical Value Table (One Tail):

T Critical Value Table (Two Tails)

References:

From Wikipedia, the free encyclopedia – statistical test (z test) – calculate the standard score – examples – standard deviation of the scores – Next, calculate the z-score

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Critical value calculator, what is critical value .

Critical Value is a statistical concept used to determine the significance of a result. It is the value of a statistic that is used to determine whether a hypothesis is true or false. It is calculated by taking the probability of a given statistic and comparing it to a predetermined level of significance. If the probability is less than the predetermined level of significance, then the hypothesis is accepted as true.

What is Critical Value Calculator ?

A Critical Value Calculator is a tool used to calculate the critical value of a statistical test. The critical value is the point at which a statistical test will reject the null hypothesis, and is determined by the level of significance and the degrees of freedom. The calculator can be used to determine the critical value for a variety of statistical tests, such as the t-test, chi-square test, and F-test.

How to Calculate Critical Value ?

The critical value is the value of a statistic that is used to determine whether a hypothesis test result is statistically significant. It is the point on the test statistic distribution at which the null hypothesis is rejected. To calculate the critical value, you need to know the degrees of freedom, the level of significance, and the type of test statistic. For example, if you are conducting a two-tailed t-test with a level of significance of 0.05 and 20 degrees of freedom, the critical value would be 2.093.

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Making statistics intuitive

Critical Value: Definition, Finding & Calculator

By Jim Frost 2 Comments

What is a Critical Value?

A critical value defines regions in the sampling distribution of a test statistic. These values play a role in both hypothesis tests and confidence intervals. In hypothesis tests, critical values determine whether the results are statistically significant. For confidence intervals, they help calculate the upper and lower limits.

In both cases, critical values account for uncertainty in sample data you’re using to make inferences about a population . They answer the following questions:

• How different does the sample estimate need to be from the null hypothesis to be statistically significant?
• What is the margin of error (confidence interval) around the sample estimate of the population parameter ?

In this post, I’ll show you how to find critical values, use them to determine statistical significance, and use them to construct confidence intervals. I also include a critical value calculator at the end of this article so you can apply what you learn.

Because most people start learning with the z-test and its test statistic, the z-score, I’ll use them for the examples throughout this post. However, I provide links with detailed information for other types of tests and sampling distributions.

Related posts : Sampling Distributions and Test Statistics

Using a Critical Value to Determine Statistical Significance

In this context, the sampling distribution of a test statistic defines the probability for ranges of values. The significance level (α) specifies the probability that corresponds with the critical value within the distribution. Let’s work through an example for a z-test.

The z-test uses the z test statistic. For this test, the z-distribution finds probabilities for ranges of z-scores under the assumption that the null hypothesis is true. For a z-test, the null z-score is zero, which is at the central peak of the sampling distribution. This sampling distribution centers on the null hypothesis value, and the critical values mark the minimum distance from the null hypothesis required for statistical significance.

Critical values depend on your significance level and whether you’re performing a one- or two-sided hypothesis. For these examples, I’ll use a significance level of 0.05. This value defines how improbable the test statistic must be to be significant.

Related posts : Significance Levels and P-values and Z-scores

Two-Sided Tests

Two-sided hypothesis tests have two rejection regions. Consequently, you’ll need two critical values that define them. Because there are two rejection regions, we must split our significance level in half. Each rejection region has a probability of α / 2, making the total likelihood for both areas equal the significance level.

The probability plot below displays the critical values and the rejection regions for a two-sided z-test with a significance level of 0.05. When the z-score is ≤ -1.96 or ≥ 1.96, it exceeds the cutoff, and your results are statistically significant.

One-Sided Tests

One-tailed tests have one rejection region and, hence, only one critical value. The total α probability goes into that one side. The probability plots below display these values for right- and left-sided z-tests. These tests can detect effects in only one direction.

Related post : Understanding One-Tailed and Two-Tailed Hypothesis Tests and Effects in Statistics

Using a Critical Value to Construct Confidence Intervals

Confidence intervals use the same critical values (CVs) as the corresponding hypothesis test. The confidence level equals 1 – the significance level. Consequently, the CVs for a significance level of 0.05 produce a confidence level of 1 – 0.05 = 0.95 or 95%.

For example, to calculate the 95% confidence interval for our two-tailed z-test with a significance level of 0.05, use the CVs of -1.96 and 1.96 that we found above.

To calculate the upper and lower limits of the interval, take the positive critical value and multiply it by the standard error of the mean. Then take the sample mean and add and subtract that product from it.

• Lower Limit = Sample Mean – (CV * Standard Error of the Mean)
• Upper Limit = Sample Mean + (CV * Standard Error of the Mean)

To learn more about confidence intervals and how to construct them, read my posts about Confidence Intervals and How Confidence Intervals Work .

Related post : Standard Error of the Mean

How to Find a Critical Value

Unfortunately, the formulas for finding critical values are very complex. Typically, you don’t calculate them by hand. For the examples in this article, I’ve used statistical software to find them. However, you can also use statistical tables.

To learn how to use these critical value tables, read my articles that contain the tables and information about using them. The process for finding them is similar for the various tests. Using these tables requires knowing the correct test statistic, the significance level, the number of tails, and, in most cases, the degrees of freedom.

The following articles provide the statistical tables, explain how to use them, and visually illustrate the results.

• T distribution table
• Chi-square table

Related post : Degrees of Freedom

Critical Value Calculator

Another method for finding CVs is to use a critical value calculator, such as the one below. These calculators are handy for finding the answer, but they don’t provide the context for the results.

This calculator finds critical values for the sampling distributions of common test statistics.

For example, choose the following in the calculator:

• Z (standard normal)
• Significance level = 0.05

The calculator will display the same ±1.96 values we found earlier in this article.

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January 16, 2024 at 5:26 pm

Hello, I am currently taking statistics and am reviewing confidence intervals. I would like to know what is the equation for calculating a two-tailed test for upper and lower limits? I would like to know is there a way to calculate one and two-tailed tests without using a confidence interval calculator and can you explain further?

January 16, 2024 at 6:43 pm

If you’re talking about calculating the critical values values for a test statistic for two-tailed test, the calculations are fairly complex. Consequently, you’ll either use statistical software, an online calculator, or a statistical table to find those limits.

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What is Critical Value

In literal terms, critical value is defined as any point present on a line which dissects the graph into two equal parts. The rejection or acceptance of null hypothesis depends on the region in which the value falls. The rejection region is defined as one of the two sections that are split by the critical value. If the test value is present in the rejection region, then the null hypothesis would not have any acceptance.

Critical Value Formula

Two formulae can be used to determine the critical value. These are listed as follows.

1.  \(\mathrm{Critical Value = \dfrac{Margin\ of\ Error} {Standard\ Deviation}}\)

2.     \( \mathrm {Critical Value = \dfrac{Margin\ of\ Error} {Standard\ Error\ of\ Sample}} \)

Anyone of the two formulae listed above can be used to determine Critical Value depending on the known values.

How to calculate critical value? - steps and process

Here are the steps you need to complete for calculating the critical value

1.    Determination of Alpha

This is the first step which the user has to complete for finding out the critical value. To determine the value of Alpha level, the following formula will be used.

\( \mathrm{Alpha\ Level} = 100% - \mathrm{Confidence Interval} \)

Consider that the confidence interval is 80%. Thus, Alpha Level will be given as.

\( \mathrm{Alpha\ Level} = 100 - 80 \)

\( \mathrm{Alpha\ Level} = 20% \)

2.    Converting the Alpha Percentage Value to Decimal

The second step involves converting the value of alpha to decimal. By default, it has the percentage unit. Hence, convert it to the decimal format. In step, the value of alpha is 20%. Thus, in terms of decimals, it would be \(0.2\)

\( \alpha = 0.2 \)

3.    Divide the value of Alpha by 2

In this step, the value of alpha determined in step 2 would be divided by \(2\). In the above example, the value of alpha is \(0.2\) .

\( \textbf{Thus}, \dfrac{\alpha} {2} = \dfrac{0.2}{2} \)

\( \dfrac{\alpha}{2} = 0.1 \)

4.    Subtract the result determined in step 3 from 1

The value of α /2 = 0.1. In this step, subtract this value from 1.

\( \textbf{Thus}, 1 - \, 0.1 = 0.9 \)

Converting this decimal value to a percentage. Thus, 0.9 would be 90%. The corresponding critical value will be for a confidence interval of 90%. It would be given as:

\( \mathbf{Z = 1.645} \)

Note: To calculate t critical value, f critical value, r critical value, z critical value and chi-square critical use our advance critical values calculator. It helps to calculate the value from the Z table very quickly in real-time.

Common confidence levels and their critical values 

The common confidence levels and the corresponding critical values in the form of a table are given below.

Types of Critical Values

To get the null hypothesis, various methods are used to determine the required area. The common methods used include z tests, t scores and also chi tests. All these methods are used to determine null hypothesis. However, null hypothesis is the area between right and left tails. The right tail has positive values while the left tail has negative ones. This point is incorporated when the critical value has to be determined.

Critical Value of Z

The standard normal model is used to determine the value of Z. The graphical display of normal distribution shows that the graph is divided into two main regions. The first one is called the Central Region and the other is the Tail Region. 

• The central region includes the values of Standard Deviation. These values are derived from the mean. 
• The tail values are on the edges of the graph. These values are determined after excluding the central region. To determine the tail values, the following formula is used.

\( \mathrm {Tail Value = 1 \space - \space Central Value} \)

Assistance offered by this critical value calculator

This tool is actually very helpful for the determination of critical value. It cuts down the time needed to determine critical value. Other than that, it is very easy to use so users are able to calculate the correct results without any difficulties.

• To use the tool, enter the degrees of freedom (DF) and the value of Alpha (α). Consider that the value of DF is 12 and Alpha is 0.5. Once the values have been entered, click the calculate button to get the results.
• In accordance with these entered values, the following results would be generated. 
• T Value is 0
• Upper Probability is 0.31
• T Value Right Tailed is 0.031  
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Tool Overview: Z Critical Value Calculator

Stuck trying to interpret the results of a statistical test - specifically finding the critical values for a standard normal distribution? You've come to the right place. Our free statistics package is intended as an alternative to Minitab and other paid software. This critical value calculator generates the critical values for a standard normal distribution for a given confidence level. The critical value is the point on a statistical distribution that represents an associated probability level. It generates critical values for both a left tailed test and a two-tailed test (splitting the alpha between the left and right side of the distribution). Simply enter the requested parameters (alpha level) into the calculator and hit calculate.

What Is a Critical Value and How Do You Use It?

A critical value is a concept from statistical testing. If we are performing hypothesis testing, we will reduce our proposition down to a single pair of choices, referred to as the null hypothesis and the alternative hypothesis . The null hypothesis denotes what we will believe to be correct if our sample data fails the statistical test. As a matter of form, it should usually reflect the default state for your process (eg. expected from normal operations). The alternative hypothesis represents an atypical outcome for the process, in which case we infer that something occured. We will then identify when and from where we shall draw a sample to assess which of these two alternatives is most likely to be correct. We will calculate a test statistic for the sample which we will compare to the expected distribution of the statistic to assess the relative probability of the hypothesis being correct. Bear in mind that this entire process exists in a probabilistic universe; we cannot opine on truth but only likelihood. We will identify the most appropriate distribution for comparing this sample (see below on when to use a standard normal vs. a t distribution). The critical value is the point in that distribution at which we must accept the alternative hypothesis as being more likely.

Note: This particular calculator is designed to find the critical value for the mean of a standard normal distribution. If your sample size is small (and you're still looking at the mean), you should use the t statistic sample distribution (we have a separate t score calculator for the t critical value). Both of these assume you're comparing the mean of the sample distribution to a fixed value. If you're trying to make inferences comparing the means of two populations (eg. both are "moving targets" vs. specified at a certain level, you're going to want to use the f statistic. (We're working on a calculator for the f critical value; you can find a table in the back of any statistics text book or here .)

In the case of the Z critical value all we need to calculate the critical value is the significance level (the alpha value) for the test. The alpha value reflects the probability of incorrectly rejecting the null hypothesis. The Z critical value is consistent for a given significance level regardless of sample size and numerator degrees. Common confidence levels for academic use are .05 (95% confidence), .025 (97.5%), and .01 (99%). That being said, a wise analyst compares the benefits of the required confidence level against the costs of achieving it (eg. don't always default to alpha .05 or .01).

How To Find Critical Values of Z

This calculator is intended to replace the use of a Z value table while providing access to a wider range of possible values for you to work with. In the offline version, you use a z score table (aka a z table) to look up the critical value for the test based on your desired level of alpha. Remember to adjust the alpha value based on wether you are doing a single-tailed test or two tailed test. In this case, we can simply split the value of alpha in two since the standard normal distsribution is symmetric about its axis. From there, finding the critical values for your test is a matter of looking up the appropriate row and column in the table. Our critical values calculator automates this process, so all you need to do is enter your alpha value and the tool will find the critical values for you.

When to Use Standard Normal (Z) vs. Student's T distribution

This calculator requires you to have sufficiently large sample that you are comfortable the values of the mean will converge on the standard normal distribution via the central limit theorem. This genreally requires you to have 30+ observations. If you are working with a smaller sample, you should consider using the version we set up to find critical values of a t-distribution . In any event, to run the hypothesis test you compare the observed value of the statistic with the t value from the t distribution table.

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Critical Value Calculator

Introduction: The Critical Value Calculator is a tool used in statistical hypothesis testing to determine the critical value corresponding to a given significance level (α) and degrees of freedom. This calculator simplifies the process of finding critical values, aiding researchers in making informed decisions based on statistical tests.

Formula: The critical value is calculated based on the significance level (α) and degrees of freedom. It represents the threshold beyond which the null hypothesis is rejected in favor of the alternative hypothesis. The calculation involves statistical methods or reference tables specific to the distribution being analyzed.

How to Use:

• Enter the significance level (α) between 0 and 1.
• Enter the degrees of freedom.
• Click the “Calculate” button.
• The critical value will be displayed in the output field.

Example: For instance, with α = 0.05 and degrees of freedom = 10, clicking calculate may yield a critical value of “1.96” (placeholder value for demonstration).

• Q: What is the significance level (α)? A: The significance level (α) is the probability of rejecting the null hypothesis when it is true.
• Q: How does the degrees of freedom affect critical values? A: Degrees of freedom impact the shape of the distribution and influence critical values.
• Q: Can I use this calculator for any distribution? A: The calculator is applicable to certain distributions, such as the t-distribution or chi-square distribution, depending on the context.
• Q: Why is it important to choose the correct significance level? A: The significance level determines the threshold for making a decision about the null hypothesis.
• Q: Are critical values symmetric around the null hypothesis value? A: For symmetric distributions, critical values are symmetric around the null hypothesis value.
• Q: What happens if the calculated test statistic exceeds the critical value? A: If the test statistic exceeds the critical value, the null hypothesis is rejected.
• Q: Can I use this calculator for one-tailed tests? A: Yes, by adjusting the significance level and direction of the test, the calculator can be used for one-tailed tests.
• Q: How do I interpret critical values in hypothesis testing? A: Critical values represent the boundaries for decision-making; if the test statistic exceeds the critical value, the null hypothesis is rejected.
• Q: How does sample size relate to critical values? A: Larger sample sizes may lead to narrower confidence intervals and more precise critical values.
• Q: Are critical values constant across different sample sizes? A: Critical values may vary with sample size, especially in the case of t-distributions.

Conclusion: The Critical Value Calculator is a valuable resource for researchers conducting hypothesis tests. By providing a quick and easy way to calculate critical values, this tool enhances the efficiency and accuracy of statistical analyses, contributing to robust and reliable research findings.

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Calculate Critical Z Value

Enter a probability value between zero and one to calculate critical value. Critical values determine what probability a particular variable will have when a sampling distribution is normal or close to normal.

Probability (p): p = 1 - α/2.

Critical Value: Definition and Significance in the Real World

- Guide Authored by Corin B. Arenas , published on October 4, 2019

Ever wondered if election surveys are accurate? How about statistics on housing, health care, and testing scores?  

In this section, we’ll discuss how sample data is tested for accuracy. Read on to learn more about critical value, how it’s used in statistics, and its significance in social science research.

What is a Critical Value?

In testing statistics, a critical value is a factor that determines the margin of error in a distribution graph.

According to Statistics How To , a site headed by math educator Stephanie Glen, if the absolute value of a test statistic is greater than the critical value, then there is statistical significance that rejects an accepted hypothesis.

Critical values divide a distribution graph into sections which indicate ‘rejection regions.’ Basically, if a test value falls within a rejection region, it means an accepted hypothesis (referred to as a null hypothesis) must be rejected. And if the test value falls within the accepted range, the null hypothesis cannot be rejected.

Testing sample data involves validating research and surveys like voting habits, SAT scores, body fat percentage, blood pressure, and all sorts of population data.

Hypothesis Testing and the Distribution Curve

Hypothesis tests check if your data was taken from a sample population that adheres to a hypothesized probability distribution. It is characterized by a null hypothesis and an alternative hypothesis.

In hypothesis testing, a critical value is a point on a distribution graph that is analyzed alongside a test statistic to confirm if a null hypothesis —a commonly accepted fact in a study which researchers aim to disprove—should be rejected.

The value of a null hypothesis implies that no statistical significance exists in a set of given observations. It is assumed to be true unless statistical evidence from an alternative hypothesis invalidates it.  

How does this relate with distribution graphs? A normal distribution curve, which is a bell-shaped curve, is a theoretical representation of how often an experiment will yield a particular result.

Elements of Normal Distribution:

• Has a mean, median, or mode. A mean is the average of numbers in a group, a median is the middle number in a list of numbers, and a mode is a number that appears most often in a set of numbers.
• 50% of the values are less than the mean
• 50% of the values are greater than the mean

Majority of the data points in normal distribution are relatively similar. A perfectly normal distribution is characterized by its symmetry, meaning half of the data observations fall on either side of the middle of the graph. This implies that they occur within a range of values with fewer outliers on the high and low points of the graph.

Given these implications, critical values do not fall within the range of common data points. Which is why when a test statistic exceeds the critical value, a null hypothesis is forfeited.

Take note: Critical values may look for a two-tailed test or one-tailed test (right-tailed or left-tailed). Depending on the data, statisticians determine which test to perform first.

Finding the Critical Value

The standard equation for the probability of a critical value is:

p = 1 – α/2

Where p is the probability and alpha (α) represents the significance or confidence level. This establishes how far off a researcher will draw the line from the null hypothesis.

The alpha functions as the alternative hypothesis. It signifies the probability of rejecting the null hypothesis when it is true. For instance, if a researcher wants to establish a significance level of 0.05, it means there is a 5% chance of finding that a difference exists.

When the sampling distribution of a data set is normal or close to normal, the critical value can be determined as a z score or t score .

Z Score or T Score: Which Should You Use?

Typically, when a sample size is big (more than 40) using z or t statistics is fine. However, while both methods compute similar results, most beginner’s textbooks on statistics use the z score.

When a sample size is small and the standard deviation of a population is unknown, the t score is used. The t score is a probability distribution that allows statisticians to perform analyses on specific data sets using normal distribution. But take note: Small samples from populations that are not approximately normal should not use the t score.

What’s a standard deviation ? This measures how numbers are spread out in a set of values, showing the amount of variation. Low standard deviation means the numbers are close to the mean set, while a high standard deviation signifies numbers are dispersed at a wider range.

Calculating Z Score

The critical value of a z score can be used to determine the margin of error, as shown in the equations below:

• Margin of error = Critical value x Standard deviation of the statistic
• Margin of error = Critical value x Standard error of the statistic

The z score , also known as the standard normal probability score, signifies how many standard deviations a statistical element is from the mean. A z score table is used in hypothesis testing to check proportions and the difference between two means. Z tables indicate what percentage of the statistics is under the curve at any given point.

The basic formula for a z score sample is:

z = (X – μ) / σ

• X is the value of the element
• μ is the population mean
• σ is the standard deviation

Let’s solve an example. For instance, let’s say you have a test score of 85. If the test has a mean (μ) of 45 and a standard deviation (σ) of 23, what’s your z score? X = 85, μ = 45, σ = 23

z = (85 – 45) / 23 = 40 / 23 z = 1.7391

For this example, your score is 1.7391 standard deviations above the mean.

What do the z scores imply?

• If a score is greater than 0, the statistic sample is greater than the mean
• If the score is less than 0, the statistic sample is less than the mean
• If a score is equal to 1, it means the sample is 1 standard deviation greater than the mean, and so on
• If a score is equal to -1, it means the sample is 1 standard deviation less than the mean, and so on

For elements in a large set:

• Around 68% fall between -1 and 1
• Around 95% fall between -2 and 2
• Around 99% fall between -3 and 3

To give you an idea, here’s how spread out statistical elements would look like under a z score graph:

Calculating T Score

On the other hand, here’s the standard formula for the t score:

t = [ x – μ ] / [ s / sqrt( n ) ]

• x is the sample mean
• s is the sample’s standard deviation
• n is the sample size

Then, we account for the degrees of freedom (df) which is the sample size minus 1. df = n – 1

T distribution, also known as the student’s distribution, is associated with a unique cumulative probability. This signifies the chance of finding a sample mean that’s less than or equal to x, based on a random sample size n. Cumulative probability refers to the likelihood that a random variable would fall within a specific range. To express the t statistic with a cumulative probability of 1 – α, statisticians use t α .

Part of finding the t score is locating the degrees of freedom (df) using the t distribution table as a reference. For demonstration purposes, let’s say you have a small sample of 5 and you want to conduct a right-tailed test. Follow the steps below.

5 df, α = 0.05

• Take your sample size and subtract 1. 5 – 1 = 4. df = 4
• For this example, let’s say the alpha level is 5% (0.05).
• Look for the df in the t distribution table along with its corresponding alpha level. You’ll find the critical value where the column and row intersect.

*One-tail t distribution table referenced from How to Statistics. In this example, 5 df, α = 0.05, the critical value is 2.132.

Here’s another example using the t score formula.

A factory produces CFL light bulbs. The owner says that CFL bulbs from their factory lasts for 160 days. Quality specialists randomly chose 20 bulbs for testing, which lasted for an average of 150 days, with a standard deviation of 40 days. If the CFL bulbs really last for 160 days, what is the probability that 20 random CFL bulbs would have an average life that’s less than 150 days?

t = [ 150 – 160 ] / [ 40 / sqrt( 20 ) ] = -10 / [40 / 4.472135] = -10 / 8.94427 t = -1.118034

The degrees of freedom: df = 20 – 1, df = 19

Again, use the variables above to refer to a t distribution table, or use a t score calculator.

For this example, the critical value is 0.1387 . Thus, if the life of a CFL light bulb is 160 days, there is a 13.87% probability that the average CFL bulb for 20 randomly chosen bulbs would be less than or equal to 150 days.

If we were to plot the critical value and shade the rejection region in a graph, it would look like this:

If a test statistic is greater than this critical value, then the null hypothesis, which is ‘CFL light bulbs have a life of 160 days,’ should be rejected. That’s if tests show more than 13.87% of the sample light bulbs (20) have a lifespan of less than or equal 150 days.

Why is Determining Critical Value Important?

Researchers often work with a sample population, which is a small percentage when they gather statistics.

Working with sample populations does not guarantee that it reflects the actual population’s results. To test if the data is representative of the actual population, researchers conduct hypothesis testing which make use of critical values.

What are Real-World Uses for It?

Validating statistical knowledge is important in the study of a wide range of fields. This includes research in social sciences such as economics, psychology, sociology, political science, and anthropology.

For one, it keeps quality management in check. This includes product testing in companies and analyzing test scores in educational institutions.

Moreover, hypothesis testing is crucial for the scientific and medical community because it is imperative for the advancement of theories and ideas.

If you’ve come across research that studies behavior, then the study likely used hypothesis testing and sampling in populations. From the public’s voting behavior, to what type of houses people tend to buy, researchers conduct distribution tests.

Studies such as how male adolescents in certain states are prone to violence, or how children of obese parents are prone to becoming obese, are other examples that use critical values in distribution testing.

In the field of health care, topics like how often diseases like measles, diphtheria, or polio occur in an area is relevant for public safety. Testing would help communities know if there are certain health conditions rising at an alarming rate. This is especially relevant now in the age of anti-vaccine activists .

The Bottom Line

Finding critical values are important for testing statistical data. It’s one of the main factors in hypothesis testing, which can validate or disprove commonly accepted information.

Proper analysis and testing of statistics help guide the public, which corrects misleading or dated information.

Hypothesis testing is useful in a wide range of disciplines, such as medicine, sociology, political science, and quality management in companies.

About the Author

Corin is an ardent researcher and writer of financial topics—studying economic trends, how they affect populations, as well as how to help consumers make wiser financial decisions. Her other feature articles can be read on Inquirer.net and Manileno.com. She holds a Master’s degree in Creative Writing from the University of the Philippines, one of the top academic institutions in the world, and a Bachelor’s in Communication Arts from Miriam College.

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t-test Calculator

Table of contents

Welcome to our t-test calculator! Here you can not only easily perform one-sample t-tests , but also two-sample t-tests , as well as paired t-tests .

Do you prefer to find the p-value from t-test, or would you rather find the t-test critical values? Well, this t-test calculator can do both! 😊

What does a t-test tell you? Take a look at the text below, where we explain what actually gets tested when various types of t-tests are performed. Also, we explain when to use t-tests (in particular, whether to use the z-test vs. t-test) and what assumptions your data should satisfy for the results of a t-test to be valid. If you've ever wanted to know how to do a t-test by hand, we provide the necessary t-test formula, as well as tell you how to determine the number of degrees of freedom in a t-test.

When to use a t-test?

A t-test is one of the most popular statistical tests for location , i.e., it deals with the population(s) mean value(s).

There are different types of t-tests that you can perform:

• A one-sample t-test;
• A two-sample t-test; and
• A paired t-test.

In the next section , we explain when to use which. Remember that a t-test can only be used for one or two groups . If you need to compare three (or more) means, use the analysis of variance ( ANOVA ) method.

The t-test is a parametric test, meaning that your data has to fulfill some assumptions :

• The data points are independent; AND
• The data, at least approximately, follow a normal distribution .

If your sample doesn't fit these assumptions, you can resort to nonparametric alternatives. Visit our Mann–Whitney U test calculator or the Wilcoxon rank-sum test calculator to learn more. Other possibilities include the Wilcoxon signed-rank test or the sign test.

Which t-test?

Your choice of t-test depends on whether you are studying one group or two groups:

One sample t-test

Choose the one-sample t-test to check if the mean of a population is equal to some pre-set hypothesized value .

The average volume of a drink sold in 0.33 l cans — is it really equal to 330 ml?

The average weight of people from a specific city — is it different from the national average?

Two-sample t-test

Choose the two-sample t-test to check if the difference between the means of two populations is equal to some pre-determined value when the two samples have been chosen independently of each other.

In particular, you can use this test to check whether the two groups are different from one another .

The average difference in weight gain in two groups of people: one group was on a high-carb diet and the other on a high-fat diet.

The average difference in the results of a math test from students at two different universities.

This test is sometimes referred to as an independent samples t-test , or an unpaired samples t-test .

Paired t-test

A paired t-test is used to investigate the change in the mean of a population before and after some experimental intervention , based on a paired sample, i.e., when each subject has been measured twice: before and after treatment.

In particular, you can use this test to check whether, on average, the treatment has had any effect on the population .

The change in student test performance before and after taking a course.

The change in blood pressure in patients before and after administering some drug.

How to do a t-test?

So, you've decided which t-test to perform. These next steps will tell you how to calculate the p-value from t-test or its critical values, and then which decision to make about the null hypothesis.

Decide on the alternative hypothesis :

Use a two-tailed t-test if you only care whether the population's mean (or, in the case of two populations, the difference between the populations' means) agrees or disagrees with the pre-set value.

Use a one-tailed t-test if you want to test whether this mean (or difference in means) is greater/less than the pre-set value.

Compute your T-score value :

Formulas for the test statistic in t-tests include the sample size , as well as its mean and standard deviation . The exact formula depends on the t-test type — check the sections dedicated to each particular test for more details.

Determine the degrees of freedom for the t-test:

The degrees of freedom are the number of observations in a sample that are free to vary as we estimate statistical parameters. In the simplest case, the number of degrees of freedom equals your sample size minus the number of parameters you need to estimate . Again, the exact formula depends on the t-test you want to perform — check the sections below for details.

The degrees of freedom are essential, as they determine the distribution followed by your T-score (under the null hypothesis). If there are d degrees of freedom, then the distribution of the test statistics is the t-Student distribution with d degrees of freedom . This distribution has a shape similar to N(0,1) (bell-shaped and symmetric) but has heavier tails . If the number of degrees of freedom is large (>30), which generically happens for large samples, the t-Student distribution is practically indistinguishable from N(0,1).

💡 The t-Student distribution owes its name to William Sealy Gosset, who, in 1908, published his paper on the t-test under the pseudonym "Student". Gosset worked at the famous Guinness Brewery in Dublin, Ireland, and devised the t-test as an economical way to monitor the quality of beer. Cheers! 🍺🍺🍺

p-value from t-test

Recall that the p-value is the probability (calculated under the assumption that the null hypothesis is true) that the test statistic will produce values at least as extreme as the T-score produced for your sample . As probabilities correspond to areas under the density function, p-value from t-test can be nicely illustrated with the help of the following pictures:

The following formulae say how to calculate p-value from t-test. By cdf t,d we denote the cumulative distribution function of the t-Student distribution with d degrees of freedom:

p-value from left-tailed t-test:

p-value = cdf t,d (t score )

p-value from right-tailed t-test:

p-value = 1 − cdf t,d (t score )

p-value from two-tailed t-test:

p-value = 2 × cdf t,d (−|t score |)

or, equivalently: p-value = 2 − 2 × cdf t,d (|t score |)

However, the cdf of the t-distribution is given by a somewhat complicated formula. To find the p-value by hand, you would need to resort to statistical tables, where approximate cdf values are collected, or to specialized statistical software. Fortunately, our t-test calculator determines the p-value from t-test for you in the blink of an eye!

t-test critical values

Recall, that in the critical values approach to hypothesis testing, you need to set a significance level, α, before computing the critical values , which in turn give rise to critical regions (a.k.a. rejection regions).

Formulas for critical values employ the quantile function of t-distribution, i.e., the inverse of the cdf :

Critical value for left-tailed t-test: cdf t,d -1 (α)

critical region:

(-∞, cdf t,d -1 (α)]

Critical value for right-tailed t-test: cdf t,d -1 (1-α)

[cdf t,d -1 (1-α), ∞)

Critical values for two-tailed t-test: ±cdf t,d -1 (1-α/2)

(-∞, -cdf t,d -1 (1-α/2)] ∪ [cdf t,d -1 (1-α/2), ∞)

To decide the fate of the null hypothesis, just check if your T-score lies within the critical region:

If your T-score belongs to the critical region , reject the null hypothesis and accept the alternative hypothesis.

If your T-score is outside the critical region , then you don't have enough evidence to reject the null hypothesis.

How to use our t-test calculator

Choose the type of t-test you wish to perform:

A one-sample t-test (to test the mean of a single group against a hypothesized mean);

A two-sample t-test (to compare the means for two groups); or

A paired t-test (to check how the mean from the same group changes after some intervention).

Two-tailed;

Left-tailed; or

Right-tailed.

This t-test calculator allows you to use either the p-value approach or the critical regions approach to hypothesis testing!

Enter your T-score and the number of degrees of freedom . If you don't know them, provide some data about your sample(s): sample size, mean, and standard deviation, and our t-test calculator will compute the T-score and degrees of freedom for you .

Once all the parameters are present, the p-value, or critical region, will immediately appear underneath the t-test calculator, along with an interpretation!

One-sample t-test

The null hypothesis is that the population mean is equal to some value μ 0 \mu_0 μ 0 ​ .

The alternative hypothesis is that the population mean is:

• different from μ 0 \mu_0 μ 0 ​ ;
• smaller than μ 0 \mu_0 μ 0 ​ ; or
• greater than μ 0 \mu_0 μ 0 ​ .

One-sample t-test formula :

• μ 0 \mu_0 μ 0 ​ — Mean postulated in the null hypothesis;
• n n n — Sample size;
• x ˉ \bar{x} x ˉ — Sample mean; and
• s s s — Sample standard deviation.

Number of degrees of freedom in t-test (one-sample) = n − 1 n-1 n − 1 .

The null hypothesis is that the actual difference between these groups' means, μ 1 \mu_1 μ 1 ​ , and μ 2 \mu_2 μ 2 ​ , is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the difference μ 1 − μ 2 \mu_1 - \mu_2 μ 1 ​ − μ 2 ​ is:

• Different from Δ \Delta Δ ;
• Smaller than Δ \Delta Δ ; or
• Greater than Δ \Delta Δ .

In particular, if this pre-determined difference is zero ( Δ = 0 \Delta = 0 Δ = 0 ):

The null hypothesis is that the population means are equal.

The alternate hypothesis is that the population means are:

• μ 1 \mu_1 μ 1 ​ and μ 2 \mu_2 μ 2 ​ are different from one another;
• μ 1 \mu_1 μ 1 ​ is smaller than μ 2 \mu_2 μ 2 ​ ; and
• μ 1 \mu_1 μ 1 ​ is greater than μ 2 \mu_2 μ 2 ​ .

Formally, to perform a t-test, we should additionally assume that the variances of the two populations are equal (this assumption is called the homogeneity of variance ).

There is a version of a t-test that can be applied without the assumption of homogeneity of variance: it is called a Welch's t-test . For your convenience, we describe both versions.

Two-sample t-test if variances are equal

Use this test if you know that the two populations' variances are the same (or very similar).

Two-sample t-test formula (with equal variances) :

where s p s_p s p ​ is the so-called pooled standard deviation , which we compute as:

• Δ \Delta Δ — Mean difference postulated in the null hypothesis;
• n 1 n_1 n 1 ​ — First sample size;
• x ˉ 1 \bar{x}_1 x ˉ 1 ​ — Mean for the first sample;
• s 1 s_1 s 1 ​ — Standard deviation in the first sample;
• n 2 n_2 n 2 ​ — Second sample size;
• x ˉ 2 \bar{x}_2 x ˉ 2 ​ — Mean for the second sample; and
• s 2 s_2 s 2 ​ — Standard deviation in the second sample.

Number of degrees of freedom in t-test (two samples, equal variances) = n 1 + n 2 − 2 n_1 + n_2 - 2 n 1 ​ + n 2 ​ − 2 .

Two-sample t-test if variances are unequal (Welch's t-test)

Use this test if the variances of your populations are different.

Two-sample Welch's t-test formula if variances are unequal:

• s 1 s_1 s 1 ​ — Standard deviation in the first sample;
• s 2 s_2 s 2 ​ — Standard deviation in the second sample.

The number of degrees of freedom in a Welch's t-test (two-sample t-test with unequal variances) is very difficult to count. We can approximate it with the help of the following Satterthwaite formula :

Alternatively, you can take the smaller of n 1 − 1 n_1 - 1 n 1 ​ − 1 and n 2 − 1 n_2 - 1 n 2 ​ − 1 as a conservative estimate for the number of degrees of freedom.

🔎 The Satterthwaite formula for the degrees of freedom can be rewritten as a scaled weighted harmonic mean of the degrees of freedom of the respective samples: n 1 − 1 n_1 - 1 n 1 ​ − 1 and n 2 − 1 n_2 - 1 n 2 ​ − 1 , and the weights are proportional to the standard deviations of the corresponding samples.

As we commonly perform a paired t-test when we have data about the same subjects measured twice (before and after some treatment), let us adopt the convention of referring to the samples as the pre-group and post-group.

The null hypothesis is that the true difference between the means of pre- and post-populations is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the actual difference between these means is:

Typically, this pre-determined difference is zero. We can then reformulate the hypotheses as follows:

The null hypothesis is that the pre- and post-means are the same, i.e., the treatment has no impact on the population .

The alternative hypothesis:

• The pre- and post-means are different from one another (treatment has some effect);
• The pre-mean is smaller than the post-mean (treatment increases the result); or
• The pre-mean is greater than the post-mean (treatment decreases the result).

Paired t-test formula

In fact, a paired t-test is technically the same as a one-sample t-test! Let us see why it is so. Let x 1 , . . . , x n x_1, ... , x_n x 1 ​ , ... , x n ​ be the pre observations and y 1 , . . . , y n y_1, ... , y_n y 1 ​ , ... , y n ​ the respective post observations. That is, x i , y i x_i, y_i x i ​ , y i ​ are the before and after measurements of the i -th subject.

For each subject, compute the difference, d i : = x i − y i d_i := x_i - y_i d i ​ := x i ​ − y i ​ . All that happens next is just a one-sample t-test performed on the sample of differences d 1 , . . . , d n d_1, ... , d_n d 1 ​ , ... , d n ​ . Take a look at the formula for the T-score :

Δ \Delta Δ — Mean difference postulated in the null hypothesis;

n n n — Size of the sample of differences, i.e., the number of pairs;

x ˉ \bar{x} x ˉ — Mean of the sample of differences; and

s s s  — Standard deviation of the sample of differences.

Number of degrees of freedom in t-test (paired): n − 1 n - 1 n − 1

t-test vs Z-test

We use a Z-test when we want to test the population mean of a normally distributed dataset, which has a known population variance . If the number of degrees of freedom is large, then the t-Student distribution is very close to N(0,1).

Hence, if there are many data points (at least 30), you may swap a t-test for a Z-test, and the results will be almost identical. However, for small samples with unknown variance, remember to use the t-test because, in such cases, the t-Student distribution differs significantly from the N(0,1)!

🙋 Have you concluded you need to perform the z-test? Head straight to our z-test calculator !

What is a t-test?

A t-test is a widely used statistical test that analyzes the means of one or two groups of data. For instance, a t-test is performed on medical data to determine whether a new drug really helps.

What are different types of t-tests?

Different types of t-tests are:

• One-sample t-test;
• Two-sample t-test; and
• Paired t-test.

How to find the t value in a one sample t-test?

To find the t-value:

• Subtract the null hypothesis mean from the sample mean value.
• Divide the difference by the standard deviation of the sample.
• Multiply the resultant with the square root of the sample size.

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ol{padding-top:0;}.css-63uqft ul:not(:first-child),.css-63uqft ol:not(:first-child){padding-top:4px;} Test setup

Choose test type

t-test for the population mean, μ, based on one independent sample . Null hypothesis H 0 : μ = μ 0  

Alternative hypothesis H 1

Test details

Significance level α

The probability that we reject a true H 0 (type I error).

Degrees of freedom

Calculated as sample size minus one.

Test results

• Informative

Critical Value Calculator

Introduction:.

Statistical analysis plays a crucial role in various fields, ranging from scientific research to business decision-making. When performing hypothesis tests, it is essential to determine the critical value to assess the significance of the test results accurately. The critical value acts as a threshold that helps determine whether to accept or reject the null hypothesis. In this article, we will delve into the concept of critical value and explore the significance of a critical value calculator in simplifying the calculation process.

Understanding Critical Value:

A critical value represents the cutoff point for a test statistic beyond which the null hypothesis is rejected. It is determined based on the chosen significance level, which indicates the probability of making a Type I error. By comparing the test statistic to the critical value, we can assess whether the observed data supports the alternative hypothesis or fails to provide sufficient evidence.

Critical Value Calculator:

A critical value calculator is a useful tool that automates the process of determining critical values for different statistical tests. It eliminates the need for manual calculations, saving time and reducing the chances of errors. These calculators are available as online tools or built-in functions in statistical software packages.

Using a Critical Value Calculator:

To use a critical value calculator effectively, follow these steps:

• Determine the significance level: Select the desired level of confidence, typically denoted as α, which corresponds to the probability of making a Type I error.
• Select the appropriate test statistic: Identify the statistical test relevant to your analysis, such as t-test, chi-square test, or F-test.
• Input the sample size or degrees of freedom: Depending on the test, provide the sample size or degrees of freedom required for the calculation.
• Calculate the critical value: Click the calculate button or execute the function, and the critical value will be generated.

Example of Using a Critical Value Calculator:

To illustrate the practical use of a critical value calculator, let’s consider a scenario where a researcher wants to test whether a new drug has a significant effect on reducing blood pressure. They set the significance level at α = 0.05 and perform a t-test on a sample of 50 individuals. Using the critical value calculator, they input the significance level, the test statistic (t), and the sample size. The calculator provides the critical value, which, in this case, may be 1.96.

Importance of Critical Value in Decision Making:

The critical value plays a pivotal role in decision making during hypothesis testing. By comparing the test statistic to the critical value, we can determine whether to accept or reject the null hypothesis. If the test statistic exceeds the critical value, we reject the null hypothesis in favor of the alternative hypothesis, indicating a significant finding. On the other hand, if the test statistic falls below the critical value, we fail to reject the null hypothesis, suggesting insufficient evidence to support the alternative hypothesis.

Factors Influencing Critical Value:

Several factors influence the critical value in hypothesis testing. The sample size affects the critical value, as larger sample sizes tend to have narrower critical regions. Moreover, the significance level chosen has a trade-off with the critical value. A higher significance level leads to a larger critical value, increasing the chances of rejecting the null hypothesis. Additionally, the relationship between the critical value and the test statistic varies depending on the statistical test employed.

Limitations of Critical Value Calculators:

While critical value calculators offer convenience and accuracy, it’s important to be aware of their limitations. These calculators assume specific conditions, such as normality and independence, which may not always hold in real-world scenarios. It is crucial to understand the underlying assumptions and validate their applicability before relying solely on the results generated by a critical value calculator.

Conclusion:

In conclusion, understanding and calculating critical values are essential in hypothesis testing, allowing us to make informed decisions based on statistical evidence. Critical value calculators simplify the process by automating the calculations and providing accurate results. By utilizing these tools, researchers and analysts can save time and ensure the accuracy of their statistical analyses.

Q1: What is the significance of the critical value in hypothesis testing? A1: The critical value helps determine whether to accept or reject the null hypothesis based on the observed test statistic.

Q2: Can I use a critical value calculator for all types of statistical tests? A2: Critical value calculators are designed for specific statistical tests and may not be applicable to all scenarios. Choose a calculator that aligns with your chosen test.

Q3: How does the sample size affect the critical value? A3: Larger sample sizes tend to result in narrower critical regions, leading to smaller critical values.

Q4: Is it necessary to specify the significance level before using a critical value calculator? A4: Yes, the significance level determines the critical value and should be defined beforehand.

Q5: Are critical value calculators always accurate? A5: Critical value calculators provide accurate results based on the assumptions and conditions they consider. However, it is important to validate the applicability of these assumptions in real-world scenarios.

F Value Calculator

No wonder, years ago, mathematicians found that when two samples from a normal population are taken, the scores of the variances of the measurements in each pair always pursue the same proportion.

Not interestingly, mathematicians have discovered that the percentage of sample variability collected in a variety of different ways follows this same distribution, the F-distribution, over the prevailing years. We can use the ratio of variances to conduct hypothesis tests because we know that sampling distributions of the ratio of variances follow a known distribution.

It is said that a critical value is a cut-off value, or two cut-off values in the case of a two-tailed test, that defines the rejection regions. In other words, critical values divide your test statistic’s scale into the rejection region and the non-rejection region.

Here’s how the critical value calculator can help you shape one and two-tailed critical values for the foremost commonly used statistical tests.

You can quickly determine the critical values for both two-tailed and one-tailed tests here. It works for the most common statistical distributions: the standard normal distribution N (0, 1), which is when you have a Z-score, T-student, chi-square, or F-distribution.

To calculate critical values, you must first understand the distribution of your test statistic under the assumption that the null hypothesis is true. The critical values are the points on the distribution that have the same possibility as your test statistic and are equal to the significance level. These values are assumed to be as extreme as the critical values.

Consider the critical value to be considered as evidence against the specified null hypothesis. It is a value obtained by a distance function with a probability equal to or greater than the null hypothesis. In an error-probabilistic framework, a proper distance function based on a test statistic has the following generic form:

X (read “X bar”) is the population baseline or control arithmetic means, 0 is the observed mean/treatment group mean, and x is the standard error of the mean.

In particular, if the test is one-sided, there will be only one critical value; if it is two-sided, there will be two: one to the left and one to the right of the distribution’s median value.

When shaping your overall results you must use the F-statistic in conjunction with the p-value. A significant result does not imply that all of your variables are significant. The statistic is simply comparing the combined effect of all the variables.

For example, if you are performing regression analysis with the F-Statistic to determine a change in R Squared, the Coefficient of Determination, you would use the p-value to get the “big picture.”

Choose the F-distribution and enter the degrees to calculate the critical value for the F-statistic which is (n – 1).

Here is we describe that F-distributed errors are common in the analysis of variance, which is widely used in the social sciences. The distribution is also known as positive values, similar to the distribution of X2.

There is no single F-distribution to speak of, as there is with the T distribution. For each pair of degrees of freedom, a different F distribution is defined one for the numerator and one for the denominator.

In order to convert the desired probability to a critical value, the inverse cumulative PDF of the F-distribution specified by the two degrees of freedom must be calculated. There is no easy way to find a critical value of f, and while there are tables, using a calculator is now the preferred method.

When deciding to support or reject the null hypothesis, the F-statistic can be used. No wonder, you’ll see an F-value and an F-critical value in your F-test results.

Click here, if you want to use Sig Fig Calculator for free

The value you originate from your statistics is known as the F-value or F-Statistic.

The F-critical value is a specific value to which your F-value is compared.

You can reject the null hypothesis if your calculated F-value in a test is greater than your F-critical value. In an F-Test, however, the statistic is only one measure of significance. You should also think about the p-value. The F-statistic determines the p-value, which is the possibility that your results occurred by chance.

Let’s define the term “F Test”, it refers to any test that employs the F-distribution. Most of the time, when people talk about F-Test, they refer to F-Test in order to compare two different variances.

The F-statistic is used in many different tests, including regression analysis. It is possible to formulate the F-test to see if a sloping linear regression produces a significantly better result than no regression.

Assume there is a table with two critical values, and one of the F-values is at the p = 0.05 level of significance. The p = 0.01 level of significance is the second critical value of F.

Calculate your F-ratio. This has (x, y) points associated with it.

Your critical F-ratio is the point of intersection.

Your result is significant at that level of probability if the value of F you obtained is equal to or greater than this critical F-value.

The F-score is also well-known as the F-measure, a measure of a test’s accuracy in binary classification statistical analysis. To compute the F-score, it takes into account both the precision and recall of the test. With the best value of 1 and the worst value of 0, the F-score can be interpreted as a weighted average of precision and recall.

The F-test calculator provides simple answers to questions such as:

What is the significance of an F-test value?

What F-test value is required to obtain a significant result?

In the end, the t-test, at least in the form presented in this text, requires that samples come from populations with equal variables; the F-test for equivalence of variables is sometimes used before the t-test for equality of means. When the stakes are high or the researcher is a little obsessive, it will be used in conjunction with t-tests.

How to Calculate Critical Value

Delving into the world of statistics, understanding how to calculate critical value is pivotal for anyone seeking accurate and reliable results in hypothesis testing. In this comprehensive guide, we will unravel the intricacies of critical values, empowering you to make informed decisions based on sound statistical principles.

Unveiling the Mystery: How to Calculate Critical Value

The significance of critical value.

Critical value, a fundamental concept in statistics, dictates the acceptance or rejection of a null hypothesis. It serves as the threshold for statistical significance, guiding researchers and analysts in drawing meaningful conclusions from their data.

Understanding the Formula

To calculate the critical value, one must consider the significance level (alpha) and the degrees of freedom. The formula varies based on the statistical test employed, whether it be a t-test, z-test, or chi-square test. Familiarizing yourself with these formulas is essential for accurate computations.

Step-by-Step Guide

1. identify the statistical test.

Different tests demand different approaches. Determine whether you’re dealing with a t-test, z-test, or another statistical method to choose the correct critical value table.

2. Set the Significance Level

Establish the alpha level, typically denoted as 0.05 or 0.01. This represents the probability of making a Type I error, and adjusting it allows you to control the sensitivity of your test.

3. Find Degrees of Freedom

Degrees of freedom vary based on the nature of your data and the statistical test. Locate the appropriate row and column in the critical value table to pinpoint the critical value.

4. Critical Value Table

Consulting a critical value table specific to your statistical test is crucial. These tables provide precise values corresponding to different alpha levels and degrees of freedom.

Advantages of Mastering Critical Value Calculation

Understanding how to calculate critical value bestows several advantages. It enhances the credibility of your research, ensures accurate hypothesis testing, and enables you to make data-driven decisions with confidence.

Navigating Challenges: Common Pitfalls in Critical Value Calculation

Misinterpretation of alpha.

Failing to grasp the significance of the alpha level can lead to misguided conclusions. A thorough understanding of its role is paramount for precise critical value determination.

Inadequate Knowledge of Degrees of Freedom

Degrees of freedom vary across statistical tests, and miscalculating them can result in erroneous critical values. Take time to familiarize yourself with the specific requirements of your chosen test.

How to Calculate Critical Value: FAQs

Q: Can critical values be negative? Critical values are inherently positive. They represent the thresholds beyond which you reject the null hypothesis, emphasizing the positive nature of these values in hypothesis testing.

Q: How does the critical value relate to p-value? While critical values and p-values are interconnected, they serve distinct roles. Critical values determine the threshold for rejection, whereas p-values indicate the probability of observing results as extreme as the ones obtained.

Q: Is there a universal critical value table? No, critical value tables vary depending on the statistical test and the degrees of freedom. It’s crucial to use a table specific to your chosen method.

Q: Can critical values change based on sample size? Critical values are influenced by the sample size, particularly in t-tests. As sample size increases, critical values tend to approach those of a standard normal distribution.

Q: Are critical values constant across different significance levels? No, critical values change with the chosen significance level. Higher significance levels result in more conservative critical values, influencing the outcome of hypothesis testing.

Q: How frequently should critical values be recalibrated? Critical values should be recalibrated whenever there are changes in the study design, sample size, or chosen significance level. Regular recalibration ensures the accuracy of hypothesis testing.

Navigating the realm of statistics becomes more manageable when armed with the knowledge of how to calculate critical value. This guide has equipped you with a comprehensive understanding of critical values, from their significance to practical applications. Mastering this skill enhances the rigor of your statistical analyses, fostering a data-driven approach to decision-making.

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Critical Value Calculators

Critical values are crucial in the hypothesis testing process, because they determine threshold points that indicate whether or not the null hypothesis should be rejected.

In our site you will find critical value calculators for the most common distributions, including critical z-scores, critical t-scores, as well as critical values for the F-distribution and the Chi-Square distributions.

These are the critical value calculators that we have available for you. Check the list below:

Critical Chi-Square Values

Critical Correlation Calculator

Critical F-Values Calculator

Critical T-values

Critical Z-Values

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