geometric mean homework answers

Geometric Mean: Definition, Examples, Formula, Uses

Statistics Definitions > Geometric Mean

• How to Find the Geometric Mean.
• Technology Options.
• Real Life Uses.

More Advanced Info:

• Equal Ratios and a Geometric Explanation.
• Logarithmic Values and Dealing with Negative Numbers.

Arithmetic Mean-Geometric Mean Inequality

Antilog of a geometric mean, geometric mean definition and formula.

Watch the video for three examples of how to find the geometric mean:

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The geometric mean is a type of average , usually used for growth rates , like population growth or interest rates. While the arithmetic mean adds items, the geometric mean multiplies items. Also, you can only get the geometric mean for positive numbers.

What this formula is saying in English is: multiply your items together and then take the nth root (where n is the number of items).

The π symbol in the formula is product notation: It’s the mathematical notation for “product”, similar to the (probably more familiar) Σ in summation notation .

How to Find the Geometric Mean (Examples)

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Example 1 : What is the geometric mean of 2, 3, and 6? First, multiply the numbers together and then take the cubed root (because there are three numbers) = (2*3*6) 1/3 = 3.30

Note : The power of (1/3) is the same as the cubed root 3 √. To convert a nth root to this notation, just change the denominator in the fraction to whatever “n” you have. So:

• 5th root = to the (1/5) power
• 12th root = to the (1/12) power
• 99th root = to the (1/99) power.

Example 2 : What is the geometric mean of 4,8.3,9 and 17? First, multiply the numbers together and then take the 5th root (because there are 5 numbers) = (4 * 8 * 3 * 9 * 17) (1/5) = 6.81

Example 3: What is the geometric mean of 1/2, 1/4, 1/5, 9/72 and 7/4? First, multiply the numbers together and then take the 5th root: (1/2*1/4*1/5*9/72*7/4) (1/5) = 0.35.

Example 4 : The average person’s monthly salary in a certain town jumped from $2,500 to $5,000 over the course of ten years. Using the geometric mean, what is the average yearly increase?

Step 1: Find the geometric mean. (2500*5000)^(1/2) = 3535.53390593.

Step 2: Divide by 10 (to get the average increase over ten years). 3535.53390593 / 10 = 353.53.

The average increase (according to the GM) is 353.53.

Tip: The geometric mean of a set of data is always less than the arithmetic mean with one exception: if all members of the data set are the same (i.e. 2, 2, 2, 2, 2, 2, then the two means are equal.

A More Detailed Example

What the answer of 1.51 is telling you is that if you multiplied your initial investment by 1.51 each year, you would get the same amount as if you had multiplied it by 1.5, 1.2 and 1.9.

• Art work value year 0: $90,000.
• Art work value year 1: $90,000 * 1.5 = $135,000
• Art work value year 2: $135,000 * 1.2 = $162,000
• Art work value year 3: $90,000 * 1.9 = $307,000

or, using the geometric mean: $90,000 * 1.50663725458 3 = $307,000*

*If you do this calculation, it’s slightly different because of the number of decimal places I wrote here. In other words, you should get the exact result on a calculator.

Back to Top

Why not use the Arithmetic Mean Instead?

The arithmetic mean is the sum of the data items divided by the number of items in the set: (1.5+1.2+1.9)/3 = 1.53

As you can probably tell, adding 1.53 to your initial price won’t get you anywhere, and multiplying it will give you the wrong result.

$90,000 * 1.53 * 1.53 * 1.53 = $322,343.91

Technology Options for Calculating the GM

• JMP does not have a formula, but you can create one with the formula editor.
• SAS : If you use SAS/STAT 12.1 or later, specify the ALLGEO statistic keyword in the PROC SURVEYMEANS statement. Earlier editions lack this capability.
• Excel : use the GEOMEAN function for any range of positive data. The syntax is GEOMEAN(number1, [number2], …)
• SPSS : Use the MEANS command. In the SPSS menus, choose Analyze>Compare Means>Means, then click on the “Options” button and select from the list of available statistics on the left.
• MINITAB : use the GMEAN function. The syntax is GMEAN(number), where “number” is the column number. All numbers must be positive.
• MAPLE : The calling sequence GeometricMean has a variety of options for calculating the mean from a data sample (A), a matrix data set(M) or a random variable or distribution(X). See this article for the full parameters.
• TI83 : there is no built in function. As a workaround, enter your data into a list and then enter the geometric mean formula on the home screen.
• TI89 : like the TI83, there isn’t a built in function. You could install an app, like the “ Statistics and Probability Made Easy ” app, which I recommend as I used it in grad school :).

Real Life Uses

Aspect ratios.

The geometric mean has been used in film and video to choose aspect ratios (the proportion of the width to the height of a screen or image). It’s used to find a compromise between two aspect ratios, distorting or cropping both ratios equally.

Computer Science

Computers use mind-boggling amounts of data which often has to be summarized using statistics. One study compared the precision of several statistics (arithmetic means, geometric means, and percentage in the top x%) for a mind-boggling 97 trillion pieces of citation data. The study found that the geometric mean was the most precise (see this Cornell University Library article ).

1. Mean Proportional The geometric mean is used as a proportion in geometry (and is sometimes called the “mean proportional”). The mean proportional of two positive numbers a and b, is the positive number x, so that:

The Geometric Mean has many applications in medicine. It has been called the “gold standard” for some measurements, including for the calculation of gastric emptying times JNM .

Proportional Growth

As the examples at the beginning of this article demonstrate, the geometric mean is useful for calculating proportional growth, like the growth seen in long term investments using the Treasury bill rate as your riskfree rate (the riskfree rate is the theoretical return rate on a risk free investment). According to NYU corporate finance and valuation professor Aswath Damodoran , the geometric mean is appropriate for estimating expected returns over long term horizons. For short terms, the arithmetic mean is more appropriate.

It’s use isn’t limited to financial markets—it can be applied anywhere there is some type of proportional growth. For example, let’s say the amount of cells in a culture are 100, 180, 210, and 300 over a four day period. This gives a growth of 1.8 for day 2, 1.167 for day 3, and 1.42 for day 4. The geometric mean is (1.8 * 1.167 * 1.42) (1/3) = 1.44, meaning a daily growth of .44 or 44%.

United Nations Human Development Index

The Human Development Index (HDI) is an index that takes into account factors other than economic development when reporting a country’s growth. It is “…a summary measure of average achievement in key dimensions of human development: a long and healthy life, being knowledgeable and have a decent standard of living. The HDI is the geometric mean of normalized indices for each of the three [categories].” UNDP The “normalized” indexes refer to the fact that the geometric mean isn’t affected by differences in scoring indices. For example, if standard of living is scored on a scale of 1 to 5 and longevity on a scale of 1 to 100, a country that scores better in longevity would score better overall if the arithmetic mean were used. The geometric mean isn’t affected by those factors.

Water Quality Standards

Test results for water quality (specifically, fecal coliform bacteria concentrations) are sometimes reported as geometric means. Water authorities identify a threshold geometric mean where beaches or shellfish beds must be closed. According to CA.GOV , the dampening effect of the geometric mean is especially useful in water quality calculations, as bacteria levels can vary from 10 to 10,000 fold over a period of time. Back to Top

Equal Ratios

When you look at the geometric mean and the numbers you put into the calculation, an interesting thing happens. Let’s say you wanted to find the geometric mean of 4 and 9. The calculation would be √(4 * 9) = 6.

• The ratio of the first number (4) and the geometric mean (6) is 4/6, which reduces to 2/3.
• The ratio of the second number (9) and the geometric mean (6) is 6/9, which reduces to 2/3.

As you can see, the ratios are the same. This tells you that the geometric mean is a sort of “average” of all of the multipliers you are putting into the equation. Take the numbers 2 and 18. What number could you put in the center so that the ratio of 2 (to this number) is the same as the ratio of this number to 18?

If you guessed 6, you’re right, because 2 * 3 = 6 and 6 * 3 = 18. For more complex numbers, the ratios would be difficult to work out, which is why the formula is used.

You can get the same result (6) by using the formula, so if you’re ever presented the above type of problem in a math class, just find the square root of the numbers multiplied together: √(2 * 18) = 6

A Geometric Explanation

Let’s say you had a rectangle with sides of 2″ and 18″. The perimeter of this rectangle has the same perimeter as a square with four sides of 10″ each. 10 is what you would get if you worked out the arithmetic mean : (2 + 18) / 2 = 20 / 2 = 10.

The geometric mean can also be equated to the rectangle-square scenario: the square root of the sides (i.e. √ 2*18) is the length of the sides of a square with the same area as the rectangle. A rectangle 2″ x 18″ = 36 square inches and 6″ x 6″ is also 36 square inches. 6 is what you would get if you worked out the geometric mean: √(2*18) = 6.

Logarithmic Values and the Geometric Mean

One way to think of the geometric mean is that it’s the average of logarithmic values converted back to base 10. If you’re familiar with logarithms , this can be a very intuitive way to look at it. For example, let’s say you wanted to calculate the geometric mean of 2 and 32.

Step 1: Convert the numbers to base 2 logs (you can theoretically use any base):

Step 2: Find the (arithmetic) average of the exponents in Step 1. The average of 1 and 5 is 3. We’re still working in base 2 here, so our average gives us 2 3 , which gives us the geometric mean of 2 * 2 * 2 = 8.

Dealing with Negative Numbers

Generally, you can only find the geometric mean of positive numbers. If you have negative numbers (common with investments), it’s possible to find a geometric mean, but you have to do a little math beforehand (which is not always easy!).

Example : What is the geometric mean for an investment that shows a growth in year 1 of 10 percent and a decrease the next year of 15 percent?

Step 1: Figure out the total amount of growth for the investment for each year. At the end of the first year you have 110% (or 1.1) of what you started with. The following year, you have 90% (or 0.9) of what you started year 2 with.

Step 2: Calculate the geometric mean based on the figures in Step 1: GM = √(1.1 * 0.9) = 0.99.

The geometric mean for this scenario is 0.99. Your investment is slowly losing money at a rate of about 1% per year.

The antilog of a number is raising 10 to its power. Let’s say your geometric mean is 8. Raise base 10 to 8: 10 8 The general formula is: antilog(g) = 10 g = 10 √(a*b)

Agresti A. (1990) Categorical Data Analysis. John Wiley and Sons, New York. Klein, G. (2013). The Cartoon Introduction to Statistics. Hill & Wamg. Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences , Wiley. Poonen, B. Retrieved March 3, 2023 from: http://mathcircle.berkeley.edu/BMC4/Handouts/inequal.pdf Vogt, W.P. (2005). Dictionary of Statistics & Methodology: A Nontechnical Guide for the Social Sciences . SAGE.

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Geometric Mean

The Geometric Mean is a special type of average where we multiply the numbers together and then take a square root (for two numbers), cube root (for three numbers) etc.

Example: What is the Geometric Mean of 2 and 18 ?

• First we multiply them: 2 × 18 = 36
• Then (as there are two numbers) take the square root: √36 = 6

In one line:

Geometric Mean of 2 and 18 = √(2 × 18) = 6

It is like the area is the same!

Example: What is the Geometric Mean of 10, 51.2 and 8 ?

• First we multiply them: 10 × 51.2 × 8 = 4096
• Then (as there are three numbers) take the cube root: 3 √4096 = 16

Geometric Mean = 3 √(10 × 51.2 × 8) = 16

It is like the volume is the same:

Example: What is the Geometric Mean of 1, 3, 9, 27 and 81 ?

• First we multiply them: 1 × 3 × 9 × 27 × 81 = 59049
• Then (as there are 5 numbers) take the 5th root: 5 √59049 = 9

Geometric Mean = 5 √(1 × 3 × 9 × 27 × 81) = 9

I can't show you a nice picture of this, but it is still true that:

1 × 3 × 9 × 27 × 81  =  9 × 9 × 9 × 9 × 9

Example: What is the Geometric Mean of a Molecule and a Mountain

Using scientific notation :

• A molecule of water (for example) is 0.275 × 10 -9 m
• Mount Everest (for example) is 8.8 × 10 3 m

Which is 1.6 millimeters , or about the thickness of a coin.

We could say, in a rough kind of way,

"a millimeter is half-way between a molecule and a mountain!"

Another cool one:

Example: What is the Geometric Mean of a Cell and the Earth?

• A skin cell is about 3 × 10 -8 m across
• The Earth's diameter is 1.3 × 10 7 m

A child is about 0.6 m tall! So we could say, in a rough kind of way,

"A child is half-way between a cell and the Earth"

So the geometric mean gives us a way of finding a value in between widely different values.

For n numbers: multiply them all together and then take the nth root (written n √ )

More formally, the geometric mean of n numbers a 1 to a n is:

n √(a 1 × a 2 × ... × a n )

The Geometric Mean is useful when we want to compare things with very different properties.

Example: you want to buy a new camera.

• One camera has a zoom of 200 and gets an 8 in reviews,
• The other has a zoom of 250 and gets a 6 in reviews.

Comparing using the usual arithmetic mean gives (200+8)/2 = 104 vs (250+6)/2 = 128 . The zoom is such a big number that the user rating gets lost.

But the geometric means of the two cameras are:

• √(200 × 8) = 40
• √(250 × 6) = 38.7...

So, even though the zoom is 50 bigger, the lower user rating of 6 is still important.

• Math Article

Geometric Mean

In mathematics and statistics, the summary that describes the whole data set values can be easily described with the help of measures of central tendencies. The most important measures of central tendencies are mean, median, mode and the range. Among these, the mean of the data set will provide the overall idea of the data. The mean defines the average of numbers. The different types of mean are Arithmetic Mean (AM), Geometric Mean (GM) and Harmonic Mean (HM). In this article, let us discuss the definition, formula, properties, applications, the relation between AM, GM, and HM with solved examples in detail.

Table of Contents:

• Difference Between AM and GM

Relation Between AM, GM and HM

• Applications

Geometric Mean Definition

In Mathematics, the Geometric Mean (GM) is the average value or mean which signifies the central tendency of the set of numbers by finding the product of their values. Basically, we multiply the numbers altogether and take the nth root of the multiplied numbers, where n is the total number of data values. For example: for a given set of two numbers such as 3 and 1, the geometric mean is equal to √(3×1) = √3 = 1.732.

In other words, the geometric mean is defined as the nth root of the product of n numbers. It is noted that the geometric mean is different from the arithmetic mean. Because, in arithmetic mean, we add the data values and then divide it by the total number of values. But in geometric mean, we multiply the given data values and then take the root with the radical index for the total number of data values. For example, if we have two data, take the square root, or if we have three data, then take the cube root, or else if we have four data values, then take the 4th root, and so on. 

Geometric Mean Formula

The formula to calculate the geometric mean is given below:

The Geometric Mean (G.M) of a series containing n observations is the nth root of the product of the values.

Consider, if x 1 , x 2 …. X n are the observation, then the G.M is defined as:

This can also be written as;

Where n = f 1 + f 2 +…..+ f n

It is also represented as:

For any Grouped Data, G.M can be written as;

Difference Between Arithmetic Mean and Geometric Mean

To learn the relation between the AM, GM and HM, first we need to know the formulas of all these three types of the mean.

Assume that “x” and “y” are the two number and the number of values = 2, then 

AM = (a+b)/2

⇒ 1/AM = 2/(a+b) ……. (1)

GM = √ (ab)

⇒GM 2 = ab ……. (2)

HM= 2/[(1/a) + (1/b)]

⇒HM = 2/[(a+b)/ab

⇒ HM = 2ab/(a+b) ….. (3)

Now, substitute (1) and (2) in (3), we get

HM = GM 2  /AM

⇒GM 2  = AM × HM

GM = √[ AM × HM]

Hence, the relation between AM, GM and HM is GM 2  = AM × HM

Geometric Mean Properties

Some of the important properties of the G.M are:

• The G.M for the given data set is always less than the arithmetic mean for the data set
• If each object in the data set is substituted by the G.M, then the product of the objects remains unchanged.
• The ratio of the corresponding observations of the G.M in two series is equal to the ratio of their geometric means
• The products of the corresponding items of the G.M in two series are equal to the product of their geometric mean.

Application of Geometric Mean

The greatest assumption of the G.M is that data can be really interpreted as a scaling factor. Before that, we have to know when to use the G.M. The answer to this is, it should be only applied to positive values and often used for the set of numbers whose values are exponential in nature and whose values are meant to be multiplied together. This means that there will be no zero value and negative value which we cannot really apply. Geometric mean has a lot of advantages and it is used in many fields. Some of the applications are as follows:

• It is used in stock indexes. Because many of the value line indexes which is used by financial departments use G.M.
• It is used to calculate the annual return on the portfolio.
• It is used in finance to find the average growth rates which are also referred to the compounded annual growth rate.
• It is also used in studies like cell division and bacterial growth etc.

Geometric Mean Examples

Here you are provided with geometric mean examples as follows

Question 1: Find the G.M of the values 10, 25, 5, and 30

Solution : Given 10, 25, 5, 30

We know that,

Therefore, the geometric mean = 13.915

Question 2 : Find the geometric mean of the following data.

Solution: Here n=5

  \(\begin{array}{l}GM =Antilog\frac{\sum \log x_{i}}{n}\end{array} \)

= Antilog 8.925/5

= Antilog 1.785

Therefore the G.M of the given data is 60.95

Question 3: Find the geometric mean of the following grouped data for the frequency distribution of weights.

From the given data, n = 160

We know that the G.M for the grouped data is

GM = Antilog ( 324.2 /160 )

GM = Antilog ( 2.02625 )

GM = 106.23

Frequently Asked Questions on Geometric Mean

What is the difference between the arithmetic mean and geometric mean.

The arithmetic mean is defined as the ratio of the sum of given values to the total number of values. Whereas in geometric mean, we multiply the “n” number of values and then take the nth root of the product.

Describe the accuracy of the arithmetic mean and geometric mean.

The geometric mean is more accurate and effective when there is more volatility in the data set. The arithmetic mean will give a more accurate answer, when the data sets independent and not skewed.

Calculate the geometric mean of 2 and 8

Let a = 2 and b = 8 Here, the number of terms, n = 2 If n =2, then the formula for geometric mean = √(ab) Therefore, GM = √(2×8) GM =√16 = 4 Therefore, the geometric mean of 2 and 8 is 4.

Mention the relation between AM, GM, and HM

The relation between AM, GM and HM is GM^2 = AM × HM. It can also be written as GM = √[ AM × HM]

If AM and HM of the data sets are 4 and 25 respectively, then find the GM.

Given that, AM = 4 HM = 25. We know that the relation between AM, GM and HM is GM = √[ AM × HM] Now, substitute AM and HM in the relation, we get; GM = √[4 × 25] GM = √100 = 10 Hence, GM = 10.

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• How to Find the Geometric Mean | Calculator & Formula

How to Find the Geometric Mean | Calculator & Formula

Published on December 2, 2021 by Pritha Bhandari . Revised on June 21, 2023.

The geometric mean is an average that multiplies all values and finds a root of the number. For a dataset with n numbers, you find the n th root of their product. You can use this descriptive statistic to summarize your data.

Table of contents

Geometric mean calculator, geometric mean formula, calculating the geometric mean, when should you use the geometric mean, example: geometric mean of percentages, example: geometric mean of widely varying values, geometric mean vs. arithmetic mean, when is the geometric mean better than the arithmetic mean, other interesting articles, frequently asked questions about central tendency, here's why students love scribbr's proofreading services.

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The geometric mean formula can be written in two ways, but they are equivalent mathematically.

In the first formula, the geometric mean is the n th root of the product of all values.

In the second formula, the geometric mean is the product of all values raised to the power of the reciprocal of n .

These formulas are equivalent because of the laws of exponents: taking the n th root of x is exactly the same as raising x to the power of 1/ n .

There are two main steps to calculating the geometric mean:

• Multiply all values together to get their product.
• Find the n th root of the product ( n is the number of values).

Before calculating this measure of central tendency , note that:

• The geometric mean can only be found for positive values.
• If any value in the dataset is zero, the geometric mean is zero.

The geometric mean is best for reporting average inflation, percentage change, and growth rates. Because these types of data are expressed as fractions, the geometric mean is more accurate for them than the arithmetic mean.

While the arithmetic mean is appropriate for values that are independent from each other (e.g., test scores), the geometric mean is more appropriate for dependent values, percentages, fractions, or widely ranging data.

We’ll walk you through some examples showing how to find the geometric means of different types of data.

Prevent plagiarism. Run a free check.

You’re interested in the average voter turnout of the past five US elections. You’ve gathered the following data.

Step 1: Multiply all values together to get their product.

Step 2: Find the n th root of the product ( n is the number of values).

The average voter turnout of the past five US elections was 54.64%.

You compare the efficiency of two machines for three procedures that are assessed on different scales. To find the mean efficiency of each machine, you find the geometric and arithmetic means of their procedure rating scores.

Geometric mean of Machine A

Geometric mean of machine b, comparing the means.

Now you compare machine efficiency using arithmetic and geometric means.

While the arithmetic means show higher efficiency for Machine B, the geometric means show that Machine B is more efficient.

The geometric mean is more accurate here because the arithmetic mean is skewed towards values that are higher than most of your dataset.

The geometric mean is more accurate than the arithmetic mean for showing percentage change over time or compound interest.

For example, say you study fruit fly population growth rates. You’re interested in understanding how environmental factors change these rates.

You begin with 2 fruit flies, and every 12 days you measure the percentage increase in the population.

Each percentage change value is also converted into a growth factor that is in decimals. The growth factor includes the original value (100%), so to convert percentage increase into a growth factor, add 100 to each percentage increase and divide by 100.

First, you convert percentage change into decimals. You add 100 to each value to factor in the original amount, and divide each value by 100.

Arithmetic mean

To find the arithmetic mean, add up all values and divide this number by n .

Geometric mean

The arithmetic mean population growth factor is 4.18, while the geometric mean growth factor is 4.05.

How do we know which mean is correct?

Because they are averages, multiplying the original number of flies with the mean percentage change 3 times should give us the correct final population value for the correct mean.

• Final population value: 2 × 4.4 × 2.87 × 5.27 ≈ 133 fruit flies
• Arithmetic mean of 418%: Final population = 2 × 4.18 × 4.18 × 4.18 ≈ 157 fruit flies
• Geometric mean of 405%:  Final population = 2 × 4.05 × 4.05 × 4.05 ≈ 133 fruit flies

Only the geometric mean gives us the true number of fruit flies in the final population. It’s the most accurate mean for the growth factor.

Even though it’s less commonly used, the geometric mean is more accurate than the arithmetic mean for positively skewed data and percentages.

In a positively skewed distribution, there’s a cluster of lower scores and a spread-out tail on the right. Income distribution is a common example of a skewed dataset.

While most values tend to be low, the arithmetic mean is often pulled upward (or rightward) by high values or outliers in a positively skewed dataset.

Because the geometric mean tends to be lower than the arithmetic mean, it represents smaller values better than the arithmetic mean.

The geometric mean is most appropriate for ratio levels of measurement, where variables have a true zero and don’t take on any negative values. Negative percentage changes have to be framed positively: for instance, −8% becomes 92% of the original value.

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

• Statistical power
• Pearson correlation
• Degrees of freedom
• Statistical significance

Methodology

• Cluster sampling
• Stratified sampling
• Focus group
• Systematic review
• Ethnography
• Double-Barreled Question

Research bias

• Implicit bias
• Publication bias
• Cognitive bias
• Placebo effect
• Pygmalion effect
• Hindsight bias
• Overconfidence bias

There are two steps to calculating the geometric mean :

Before calculating the geometric mean, note that:

• If any value in the data set is zero, the geometric mean is zero.

The arithmetic mean is the most commonly used type of mean and is often referred to simply as “the mean.” While the arithmetic mean is based on adding and dividing values, the geometric mean multiplies and finds the root of values.

Even though the geometric mean is a less common measure of central tendency , it’s more accurate than the arithmetic mean for percentage change and positively skewed data. The geometric mean is often reported for financial indices and population growth rates.

Measures of central tendency help you find the middle, or the average, of a data set.

The 3 most common measures of central tendency are the mean, median and mode.

• The mode is the most frequent value.
• The median is the middle number in an ordered data set.
• The mean is the sum of all values divided by the total number of values.

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BUS204: Business Statistics

Descriptive Statistics

Read this chapter, which discusses how to describe locations within a sample and how to analyze data from a sample. Be sure to attempt the practice problems and homework at the end of each section.

Geometric Mean

Finally, we note that the formula for the geometric mean requires that all numbers be positive, greater than zero. The reason of course is that the root of a negative number is undefined for use outside of mathematical theory. There are ways to avoid this problem however. In the case of rates of return and other simple growth problems we can convert the negative values to meaningful positive equivalent values. Imagine that the annual returns for the past three years are +12%, -8%, and +2%. Using the decimal multiplier equivalents of 1.12, 0.92, and 1.02, allows us to compute a geometric mean of 1.0167. Subtracting 1 from this value gives the geometric mean of +1.67% as a net rate of population growth (or financial return). From this example we can see that the geometric mean provides us with this formula for calculating the geometric (mean) rate of return for a series of annual rates of return:

Chapter 8, Lesson 1: Geometric Mean

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Geometric Mean Calculator

Use this online calculator to easily calculate the Geometric mean for a set of numbers or percentages. Works properly with negative numbers so it can be used to find the geometric mean of returns of investments.

Related calculators

• What is a geometric mean?
• Geometric mean formula
• How to calculate Geometric mean?
• Geometric mean for negative numbers
• Geometric means with zeros in the dataset
• Example usage in Finance
• Example usage in Social Sciences
• Other applications

    What is a geometric mean?

The geometric mean, often referred to as the geometric average, is a so-called specialized average and is defined as the n-th root of the product of n numbers of the same sign . If in an arithmetic mean we combine the numbers using the summation operation and then divide by their number, in a geometric mean we calculate the product of the numbers and then take its n-th root. Any time you have several factors contributing to a product, and you want to calculate the "average" of the factors, the answer is the geometric mean.

It is useful in a number of situations where a growth rate is of interest, for example in calculating compound interest rates, financial returns or risk and loses, area and volume averages, in computing indexes such as the U.S. Consumer Price Index (index of inflation), and others. If you are dealing with such tasks, a geometric mean calculator like ours should be most helpful.

    Geometric mean formula

A geometric approach to explain the formula is through rectangles and squares. If we have a rectangle with sides 4 and 16, the perimeter of the rectangle is the sum of all four sides: 4 + 4 + 16 + 16 = 40. The arithmetic mean of 4 and 16 is 10, and a square with a side of 10 will have the same perimeter as one with sides 4 and 16.

Now, if we take the area of our 4 x 16 rectangle instead, it is the product of 4 and 16 and equals 64. The geometric mean answers the question: what side should a square have so that its area is 64? The answer is 8, which is exactly the geometric mean of 4 and 16.

In the image above the perimeter calculation corresponds to finding the arithmetic mean and the area calculation - to finding the geometric mean.

    How to calculate Geometric mean?

Assuming you don't want to use a calculator, obviously. Let's say we have a set of numbers: 1 5 10 13 30 and we want to calculate their arithmetic mean. We would just sum the numbers (1 + 5 + 10 + 13 + 30) and then divide by 5, giving us an arithmetic mean of 11.80. To calculate the geometric mean, we take their product instead: 1 x 5 x 10 x 13 x 30 = 19,500 and then calculate the 5-th root of 19,500 = 7.21. In this case finding the geometric mean is equivalent to raising 19,500 to the 1/5-th power.

Another way to calculate the geometric mean is with logarithms , as it is also the average of logarithmic values converted back to base 10. Let's say you want to calculate the geomean of 2 and 8. It is handy to use log with base 2 here, so 2 = 2 1 and 8 = 2 3 . The arithmetic mean of the exponents (1 and 3) is 2, so the geometric mean is 2 2 = 4. This can be verified by our geometric average calculator as well.

As you can see, the geometric mean is significantly more robust to outliers / extreme values. For example, replacing 30 with 100 would yield an arithmetic mean of 25.80, but a geometric mean of just 9.17, which is very desirable in certain situations. However, before settling on using the geometric mean, you should consider if it is the right statistic to use to answer your particular question.

    Geometric mean for negative numbers

From the definition we can see that we can only calculate the geometric mean of positive numbers, or, more precisely, the numbers need to be of the same sign, in order to avoid taking the root of a negative product, which would result in imaginary numbers. However, this doesn't mean we can't work with negative numbers as well. Let's say we have the following relative changes in 3 consecutive years: 8% growth, 10% decline, 11% growth. The total growth at the end is 7.89%, but how do we calculate the average yearly growth rate? 10% would usually be -10%, having a different sign and forbidding us from doing the calculation, but we can do a little trick and express the numbers as proportions, thus 8% growth becomes 1 + 8% x 1 = 1.08, 10% decline becomes 1 - 10% x 1 = 0.9 and 11% growth becomes 1 + 11% x 1 = 1.11. The geometric mean is 1.0256 which equals 2.56% average growth per year.

Our geometric mean calculator handles this automatically, so there is no need to do the above transformations manually. You can also enter the numbers with %, like "2% 10% -10% 8%" and will deal with that as well (it simply strips the %).

    Geometric means with zeros in the dataset

The geometric mean won’t be meaningful if zeros are present in the data. You may be tempted to adjust them in some way so that the calculation can be done. There are same cases when adjustments are justified and the first one is similar to the negative numbers case above. If the data is percentage increases, you can transform them into normal percentage values in the way described for negative numbers. Zeros then become 100% or 1 and the calculation proceeds as normal.

In other cases, zeros mean non-responses and in some cases they can just be deleted before calculation. Of course, this would change the meaning of the reported statistic from applying to the whole dataset to just those people who responded, or those sensors that continue working. Due to these complications, our software would not automatically adjust zeros in any way. You might need to look for another calculator if such an adjustment is desirable.

    Example usage in Finance

When you evaluate an offer for a deposit with compound interest , or the expected returns from an investment strategy, you need to use the geometric average, not the arithmetic average. Let's see a quick example: if you hold money in a mutual fund for two years and it increased the value of its shares by 10% on the first year, and lost 10% on the second year, by using the arithmetic average of (15% - 15%)/2 = 0% you would expect to be where you started, but in fact you would have lost 2.25% of your initial investment (1.15 x 0.85)^ 1 / 2 = 0.9775 or 97.75%, losing an average of 1.13% per year.

For a more complex example, let's say you are evaluating a strategy that projects the following return on investment for the next 5 years: 6%, 7%, 8%, -35%, 10%. The arithmetic average would be 0.4% return, but the actual average yearly return over those 5 years would be -2.62%, thus it will lose you money, despite having a positive return in 4 out of 5 years.

Using the arithmetic average of 0.4% growth per year we expect to see an end capital of $1020.16, with the geometric average of -2.62% we see exactly $875.83.

Due to its qualities in correctly reflecting investment growth rates the geometric mean is used in the calculation of key financial indicators such as CAGR .

    Example usage in Social Sciences

Human population growth rate is expressed as a percentage of the current populations, and thus when it needs to be averaged, finding the geometric mean is the proper calculation to do so you can say "the average rate of growth of the population of North America over the past X years was Y%".

In surveys and studies too, the geometric mean becomes relevant. For example, if a survey found that over the years, the economic status of a poor neighborhood is getting better, they need to quote the geometric mean of the development, averaged over the years in which the survey was conducted. The arithmetic mean will not make sense in this case either.

    Other applications

The geometric mean can be useful in many other situations. For example, the geometric mean is the only correct mean when averaging normalized results [1] , which are any results that are presented as ratios to a reference value or values. This is the case when presenting performance with respect to a reference baseline performance, or when computing a single average index from several heterogeneous sources, for example an index composed of indices for health adjusted life expectancy, education years, and infant mortality. In such scenarios the arithmetic or harmonic mean would change the ranking of the results depending on what is used as a reference, while the geomean would preserve them as it is indifferent to the scales used.

The geometric mean is heavily used in geometry . In a right-angled triangle, its altitude is the length of a line extending perpendicularly from the hypotenuse to its 90° vertex. Imagining that this line splits the hypotenuse into two segments, the geometric mean of these segment lengths is the length of the altitude. In another example: the distance to the horizon of a sphere is the geometric mean of the distance to the closest point of the sphere and the distance to the farthest point of the sphere.

It also played a role in deciding on the 16:9 aspect ratio in modern monitors and TV screens [2] . The geometric mean has been used in choosing a compromise aspect ratio between the 4:3 and 2.35:1 ratios as it provided a compromise between them, distorting or cropping both in some sense equally. As you can check using our geomean calculator, the geomean of 1.333(3) and 2.35 is 1.77, which is exactly the ratio between 16 and 9 used in modern 16:9 TV screens and computer monitors.

    References

1 Philip J. Fleming and John J. Wallace. 1986. How not to lie with statistics: the correct way to summarize benchmark results. Commun. ACM 29, 3 (March 1986), 218-221. [2] TECHNICAL BULLETIN: Understanding Aspect Ratios" (PDF). The CinemaSource Press. 2001. Retrieved Feb 02, 2018.

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., "Geometric Mean Calculator" , [online] Available at: https://www.gigacalculator.com/calculators/geometric-mean-calculator.php URL [Accessed Date: 07 Sep, 2024].

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