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## Geometric Mean

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. In mathematics and statistics, measures of central tendencies describe the summary of whole data set values. The most important measures of central tendencies are mean, median, mode, and range. Among these, the mean of the data set provides the overall idea of the data. The mean defines the average of numbers in the data set. The different types of mean are Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM). In this lesson, let us discuss the definition, formula, properties, applications, the relation between AM, GM, and HM with solved examples in the end.

## Geometric Mean Definition

The **Geometric Mean (GM)** is the average value or mean which signifies the central tendency of the set of numbers by taking the root of the product of their values. Basically, we multiply the ‘n’ values altogether and take out the n^{th }root of the numbers, where n is the total number of values. For example: for a given set of two numbers such as 8 and 1, the geometric mean is equal to √(8×1) = √8 = 2√2.

Thus, the geometric mean is also defined as the n^{th} root of the product of n numbers. It is to be noted that the geometric mean is different from the arithmetic mean. In the arithmetic mean, data values are added and then divided by the total number of values. But in geometric mean, the given data values are multiplied, and then you take the root with the radical index for the final product of data values. For example, if you have two data, take the square root, or if you have three data, then take the cube root, or else if you have four data values, then take the 4^{th} root, and so on.

## Geometric Mean Formula

The Geometric Mean (G.M) of a data set containing n observations is the n^{th} root of the product of the values. Consider, if x1,x2….xn are the observation, for which we aim to calculate the Geometric Mean. The formula to calculate the geometric mean is given below:

## Difference Between Arithmetic Mean and Geometric Mean

Here is a table representing the difference between arithmetic and geometric mean.

Arithmetic Mean |
Geometric Mean |

In the arithmetic mean, data values are added and then divided by the total number of values. | Geometric Mean can be found by multiplying all the numbers in the given data set and take the n^{th} root for the obtained result. |

For example, the given data sets are: 10, 15 and 20 Here, the number of data points = 4 Arithmetic mean or mean = (10+15+20)/4 Mean = 45/3 =15 |
For example, for data set, 4, 10, 16, 24 Here n = 4 Therefore, the G.M = 4 = 4th root of 15360 G.M = 11.13 |

## Relation Between AM, GM, and HM

Before we learn the relation between the AM, GM and HM, we need to know the formulas of all these 3 types of mean. Assume that “a” and “b” are the two number and the number of values = 2, then

⇒ 1/AM = 2/(a+b) ……. (I)

GM = **√**(ab)

⇒GM^{2} = ab ……. (II)

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

⇒HM = 2/[(a+b)/ab

⇒ HM = 2ab/(a+b) ….. (III)

Now, substitute (I) and (II) in (III), we get

HM = GM^{2} /AM

⇒GM^{2} = AM × HM

Or else,

GM = √[ AM × HM]

Hence, the relation between AM, GM, and HM is **GM ^{2} = AM × HM. **Therefore the square of the geometric mean is equal to the product of the arithmetic mean and the harmonic mean.

Let us also see why the G.M for the given data set is always less than the arithmetic mean for the data set. Let A and G be A.M. and G.M.

So,

A = (a+b)/2 and G=√ab

Now let’s subtract the two equations

A−G = (a+b)/2 − √ab = (a+b−2√ab)/2 = (√a−√b)^{2}/2 ≥ 0

A−G ≥ 0

**This indicates that ****A ≥ G**

## Application of Geometric Mean

Geometric mean has many advantages over arithmetic mean 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 are used by financial departments make use of G.M.
- To calculate the annual return on the investment portfolio.
- The geometric mean is used in finance to find the average growth rates which are also known as the compounded annual growth rate (CAGR).
- Geometric Mean is also used in biological studies like cell division and bacterial growth rate etc.

**☛Tips & Tricks on Geometric Mean**

Some of the tips and tricks on G.M are as follows:

- The G.M for the given data set is always less than the arithmetic mean for the data set.
- If each value in the data set is substituted by the G.M, then the product of the values 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 the two series are equal to the product of their geometric mean.

## Geometric Mean Examples

**Example 1: Find the geometric mean of 1,3,5,7,9**

**Solution :**

The geometric mean is given as (x_{1} × x_{2} × x_{3}…× x_{n})^{1/n}

= (1 × 3 × 5 × 7 × 9)^{1/5}

= (945)^{1/5}

= 3.936

**Answer:** **Therefore, the geometric mean = 13.915**

**Example 2: Find the geometric mean of the following data.**

Weight of Table (Kg) |
Log x |

45 |
1.653 |

60 |
1.778 |

48 |
1.681 |

100 |
2.000 |

65 |
1.813 |

Total |
8.925 |

**Solution:**

Solution: Here n = 5

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

Weights of Cellphones (g) |
No of Cellphones (f) |

60-80 |
22 |

80-100 |
38 |

100-120 |
45 |

120-140 |
35 |

140-160 |
20 |

Total |
160 |

**Solution:**

Weights of Cellphones (g) |
No of Cellphones (f) |
Mid x |
Log x |
f log x |

60-80 | 22 | 70 | 1.845 | 40.59 |

80-100 | 38 | 90 | 1.954 | 74.25 |

100-120 | 45 | 110 | 2.041 | 91.85 |

120-140 | 35 | 130 | 2.114 | 73.99 |

140-160 | 20 | 150 | 2.716 | 43.52 |

Total |
160 |
324.2 |

From the given data, n = 160

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

## FAQs 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 n^{th} root of the product.

### What is Geometric Mean in Statistics?

Geometric Mean is the value or mean of a set of data points which is calculated by raising the product of the points to the reciprocal of the number of the data points.

### What Is the Geometric Mean of 2 and 8?

The geometric mean of 2 and 8 can be calculated as

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.

### What Is Geometric Mean Formula?

Geometric mean is the nth root of the product of the elements in a sequence. Geometric mean formula for given set {x1, x2, x3, …, xn} is given by (x1 × x2 × x3 × … × xn)^{1/n}

### Why Is Geometric Mean Better Than Arithmetic?

The geometric mean differs from the arithmetic average, or arithmetic mean, in how it is calculated because it takes into account the compounding that occurs from period to period. Because of this, investors usually consider the geometric mean a more accurate measure of returns than the arithmetic mean.

### Example Question Using Geometric Mean Formula

**Question 1: **Find the geometric mean of 4 and 3.

**Solution: **Using the formula for G.M., the geometric mean of 4 and 3 will be:

Geometric Mean will be √(4×3)

= 2√3

So, GM = 3.46

**Question 2: **What is the geometric mean of 4, 8, 3, 9 and 17?

**Solution:**

**Step 1: **n = 5 is the total number of values. Now, find 1/n.

1/5 = 0.2.

**Step 2: **Find geometric mean using the formula:

(4 × 8 × 3 × 9 × 17)^{0.2}

So, geometric mean = 6.814

## Frequently Asked Questions

### What is the Geometric Mean of 4 and 25?

Using the formula of geometric mean,

GM of 4, 25 = √(4×25) = √100

So, the geometric mean of 4 and 25 is 10.

### How to Define Geometric Mean Formula?

Geometric mean formula is obtained by multiplying all the numbers together and taking the nth root of the product. Visit BYJU’S to learn more about the formula of geometric mean along with solved example questions.

### Why Geometric Mean is Better than Arithmetic Mean?

Both the geometric mean and arithmetic mean are used to determine the average. For any two positive unequal numbers, the geometric mean is always less than the arithmetic mean. Now, the geometric mean is better since it takes indicates the central tendency. In certain cases, arithmetic mean works better like in representing average temperatures, etc.

## What Is the Geometric Mean?

The geometric mean is the average of a set of products, the calculation of which is commonly used to determine the performance results of an investment or portfolio. It is technically defined as “the *nth* root product of *n* numbers.” The geometric mean must be used when working with percentages, which are derived from values, while the standard arithmetic mean works with the values themselves.

The geometric mean is an important tool for calculating portfolio performance for many reasons, but one of the most significant is it takes into account the effects of compounding.

- The geometric mean is the average rate of return of a set of values calculated using the products of the terms.
- Geometric mean is most appropriate for series that exhibit serial correlation—this is especially true for investment portfolios.
- Most returns in finance are correlated, including yields on bonds, stock returns, and market risk premiums.
- For volatile numbers, the geometric average provides a far more accurate measurement of the true return by taking into account year-over-year compounding that smooths the average

## Understanding the Geometric Mean

The geometric mean, sometimes referred to as compounded annual growth rate or time-weighted rate of return, is the average rate of return of a set of values calculated using the products of the terms. What does that mean? Geometric mean takes several values and multiplies them together and sets them to the 1/n^{th} power.

For example, the geometric mean calculation can be easily understood with simple numbers, such as 2 and 8. If you multiply 2 and 8, then take the square root (the ½ power since there are only 2 numbers), the answer is 4. However, when there are many numbers, it is more difficult to calculate unless a calculator or computer program is used.

The longer the time horizon, the more critical compounding becomes, and the more appropriate the use of geometric mean.

The main benefit of using the geometric mean is the actual amounts invested do not need to be known; the calculation focuses entirely on the return figures themselves and presents an “apples-to-apples” comparison when looking at two investment options over more than one time period. Geometric means will always be slightly smaller than the arithmetic mean, which is a simple average.

## How to Calculate the Geometric Mean

To calculate compounding interest using the geometric mean of an investment’s return, an investor needs to first calculate the interest in year one, which is $10,000 multiplied by 10%, or $1,000. In year two, the new principal amount is $11,000, and 10% of $11,000 is $1,100. The new principal amount is now $11,000 plus $1,100, or $12,100.

In year three, the new principal amount is $12,100, and 10% of $12,100 is $1,210. At the end of 25 years, the $10,000 turns into $108,347.06, which is $98,347.05 more than the original investment. The shortcut is to multiply the current principal by one plus the interest rate, and then raise the factor to the number of years compounded. The calculation is $10,000 × (1+0.1) 25 = $108,347.06.

## Example of Geometric Mean

If you have $10,000 and get paid 10% interest on that $10,000 every year for 25 years, the amount of interest is $1,000 every year for 25 years, or $25,000. However, this does not take the interest into consideration. That is, the calculation assumes you only get paid interest on the original $10,000, not the $1,000 added to it every year. If the investor gets paid interest on the interest, it is referred to as compounding interest, which is calculated using the geometric mean.

Using the geometric mean allows analysts to calculate the return on an investment that gets paid interest on interest. This is one reason portfolio managers advise clients to reinvest dividends and earnings.

The geometric mean is also used for present value and future value cash flow formulas. The geometric mean return is specifically used for investments that offer a compounding return. Going back to the example above, instead of only making $25,000 on a simple interest investment, the investor makes $108,347.06 on a compounding interest investment.

Simple interest or return is represented by the arithmetic mean, while compounding interest or return is represented by the geometric mean.

## How to Find the Geometric Mean (Examples)

**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?

**Solution:**

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 thanthe 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

Let’s say you own a piece of art that increases in value by 50% the first year after you buy it, 20% the second year, and 90% the third year. What these numbers tell you is that at the end of the first year the value was multiplied by 150% or 1.5, the second year the value at the end of year 1 was multiplied by 120% or 1.2 and at the end of the third year the value at the end of year 2 was multiplied by 190% or 1.9. As these are *multiplied*, what you are looking for is the geometric mean which can be calculated in the following way:

(1.5*1.2*1.9)^{(1/3)} = 1.50663725458… or about 1.51

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.

## 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).

### Geometry

**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:

If you have similar triangles, you can use the proportion to find missing sides. For example, the leg rule states that: hypotenuse/leg = leg/projection. These types of problems appear in high school geometry classes.

**2. Golden Ratio**

The golden mean has a value of about 1.618 and can be derived from the geometric mean and similar rectangles. Let’s say you have a rectangle with width “a” and length “b”. Create a square within the rectangle with sides “a”:

The smaller rectangle to the left is similar to the larger rectangle. Both rectangles contain the golden ratio, which is the ratio of the rectangle’s length to width. You could write a statement about the relationship between the two rectangles:

b : a = a : b – a

This statement, a proportional statement about the rectangles, also defines the geometric mean as “a”.

### Medicine

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.

## 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?

2 (?) 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.

Similarly, the geometric mean of three numbers a, b, and c is a cuboid with sides a, b, and c equal to the three numbers.

## 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):

- 2 = 2
^{1} - 32 = 2
^{5}

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.

## Arithmetic Mean-Geometric Mean Inequality

The Arithmetic Mean-Geometric Mean Inequality (AM-GM inequality) states that for a list of non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean. Taking the formulas for both types of mean, we get the inequality:

For example, for the sequence of numbers {9, 12, 54}, the arithmetic mean, 25, is greater than the geometric mean, 18. The multiple proofs of this inequality are beyond the scope of this statistics site, but Bjorn Poonen offers this simple one line proof for the AM-GM inequality with two variables:

## Antilog of a Geometric Mean

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)}

## Calculating the Geometric Mean | Explanation with Examples

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.

The geometric mean is an alternative to the arithmetic mean, which is often referred to simply as “the mean.” While the arithmetic mean is based on adding values, the geometric mean multiplies values.

## Calculating the geometric mean

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.

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## When should you use the geometric mean?

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.

## Example: Geometric mean of percentages

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

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

## Example: Geometric mean of widely varying values

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.

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

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.

## Geometric mean vs. arithmetic mean

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*.

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

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.

## When is the geometric mean better than the arithmetic mean?

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.

## Frequently asked questions about central tendency

**How do I calculate the geometric mean?**

There are two 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 the geometric mean, note that:

- The geometric mean can only be found for positive values.
- If any value in the data set is zero, the geometric mean is zero.

**What’s the difference between the arithmetic and geometric means?**

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.

**What are measures of central tendency?**

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.

## 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 a **Molecule and a Mountain**

Another cool one:

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

## Definition

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})

## Useful

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

**Geometric Mean Definition and Formula**

The geometric mean definition and formula given below will clear your concepts of geometric mean and help you to calculate the geometric mean for a given data.

The Geometric mean (G.M.) of a series, including n observation, is the nth root of the product of values.

For example, if y₁, y₂, y₃, …y_{n} are the observation, then the geometric formula is defined as

Or

(*x*1,*x*2,…*xn*)^{1/n}

The geometric mean formula can also be represented in the following way:

Log GM = I/n log (x₁,x₂,…x_{n})

= 1/n (log x₁ + logx₂ +…….+ log x_{n})

= Σ log x_{i }/ n

Hence, Geometric Mean, GM is equaled to

Antilog Σ log x_{i} / n

Where n = f_{1}+ f₂ +….+ f_{n}

The formula to find geometric mean can also be written as

**Geometric Mean Formula for Grouped Data**

If we have a set of n positive values with some repeated values such as x₁,x₂,x_{3}…..x_{n,} and the values are repeating s₁,s₂,s_{3}…….s_{k }times, then the geometric mean formula for grouped data is defined as.

**Properties of Geometric Mean**

The following are the properties of Geometric mean:

- The geometric mean for a given data is always less than the arithmetic means for a given data set.
- The ratio of the associated observation of the geometric mean in two series is equivalent to the ratio of their geometric means.
- The product of the associated observation of the geometric mean in two series is equivalent to the product of their geometric means.
- If the geometric mean replaces each observation in the given data set, then the product of observations does nor change.

**Solved Examples**

1. Find the geometric mean for the following data.

Solution:

= Antilog (2.4814)

= 11.958

2. Calculate the geometric mean of 10,5,15,8,12.

Solution:

Given, x₁ = 10 ,x₂= 5,x₃ = 15,x₄ = 8,x₅= 12

N= 5

Using the geometric mean formula,

3. Find the geometric mean for the following data.

Solution: Here n = 5

Geometric Mean = Σlog x_{i }/n

= Antilog 8.925/5

= Antilog 1.785

## FAQs (Frequently Asked Questions)

**1. What is the Difference Between Arithmetic Mean and Geometric Mean.**

Ans. The difference between the arithmetic mean and Geometric mean is given below in the tabulated form.

**2. What are the Adva**n**tages and Disadvantages of the Geometric Mean?**

Ans. **Some Advantages of the Geometric Mean are:**

- The geometric mean is rigidly defined as their values are always fixed.
- It is based on all the elements of the series.
- The geometric mean is not much affected by sampling fluctuations.
- Geometric means are accurate for algebraic calculations and other mathematical operations.
- It gives more weightage to small items.

**Disadvantages of Geometric Mean**

- It is not commonly used as it is difficult to understand.
- It is not easy to calculate geometric mean in case the value of the variable in the series is negative or zero.
- The value of geometric mean cannot be obtained in case of open-end distribution.

### Why use Geometric Mean?

The arithmetic mean is the calculated average of the middle value of a data series. It is accurate to take an average of independent data, but weakness exists in a continuous data series calculation.

Example: An investor has annual return of 5%, 10%, 20%, -50%, and 20%.

Using the arithmetic mean, the investor’s total return is (5%+10%+20%-50%+20%)/5 = 1%

By comparing the result with the actual data shown on the table, the investor will find a 1% return is misleading.

Year | Starting Equity | Return % | Return $ | Closing equity |

1 | $1,000 | 5% | $50 | $1,050 |

2 | $1,050 | 10% | $105 | $1,155 |

3 | $1,155 | 20% | $231 | $1,386 |

4 | $1,386 | -50% | -$693 | $693 |

5 | $693 | 20% | $138.6 | $831.6 |

The actual 5 year return on the account is ($831.6 – $1,000)/$1,000 = -16.84%

The geometric mean is used to tackle continuous data series which the arithmetic mean is unable to accurately reflect.

### Geometric Mean Formula for Investments

Geometric Mean = [Product of (1 + Rn)] ^ (1/n) -1

Where:

- Rn = growth rate for year N

Using the same example as we did for the arithmetic mean, the geometric mean calculation equals:

##### 5th Square Root of ((1 + 0.05)(1 + 0.1)(1 + 0.2)(1 – 0.5)(1 + 0.2)) – 1 = -0.03621

Multiply the result by 100 to calculate the percentage. This results in a -3.62% annual return.

### Example of the Geometric Mean in Finance

Return, or growth, is one of the important parameters used to determine the profitability of an investment, either in the present or the future. When the return or growth amount is compounded, the investor needs to use the geometric mean to calculate the final value of the investment.

Case example: an investor is offered two different investment options. The first option is a $20,000 initial deposit with a 3% interest rate for each year over 25 years. The second option is a $20,000 initial deposit, and after 25 years the investor will get $40,000. Which investment should the investor choose?

The investor will use the future value or the present value formula, which is derived from the geometric mean. Here are the formulas used to calculate each:

**Future value = E*(1+r)^n Present value = FV*(1/(1+r)^n)**

Where:

- E = Initial equity
- r = interest rate
- FV = Future value
- n = number of years

The investor will compare both investment options by analyzing the interest rate or the final equity value with the same initial equity.

Option 1 – Future value

**Future value = E*(1+r)^n**

= $20,000*(1+0.03)^25

= $20,000*2.0937

= $41,875.56

Option 2 – Present value

**Present value = FV*(1/(1+r)^n)**

$20,000 = $40,000*(1/(1+r)^25)

0.5 = (1/(1+r)^25)

0.973 = 1/(1+r)

r = 0.028 or 2.8%

From the calculation, the investor should choose option one because it is a better investment option based on the following:

It offers a better future value of $41,875.56 vs. $40,000 or a higher interest rate of 3% vs. 2.8%.

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