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.

Contents

- 1 Which is better geometric mean or arithmetic mean?
- 2 What is the difference between arithmetic and geometric mean?
- 3 Is arithmetic mean greater than geometric mean?
- 4 How is geometric mean used in real life?
- 5 What type of data is the geometric mean used?
- 6 When arithmetic mean geometric mean and harmonic mean are equal?
- 7 Is harmonic mean smaller than geometric mean?
- 8 Is harmonic mean greater than arithmetic mean?
- 9 What are the advantages of geometric mean?
- 10 Is geometric mean affected by extreme values?
- 11 Is the geometric mean of two regression coefficient?
- 12 Geometric Mean vs Arithmetic Mean
- 13 Geometric Mean vs Arithmetic Mean
- 14 Differences in Arithmetic & Geometric Mean
- 15 Formulas for Calculation
- 16 The Effect of Outliers
- 17 Uses
- 18 Arithmetic, Geometric, and Harmonic Means for Machine Learning
- 19 Tutorial Overview
- 20 What Is the Average?
- 21 Arithmetic Mean
- 22 Geometric Mean
- 23 Harmonic Mean
- 24 How to Choose the Correct Mean?
- 25 Further Reading
- 26 Summary
- 27 Get a Handle on Statistics for Machine Learning!
- 28 Question Corner – Applications of the Geometric Mean
- 29 Arithmetic vs Geometric Mean: Which to use in Performance Appraisal
- 30 Geometric Mean: Definition, Examples, Formula, Uses
- 31 Geometric Mean Definition and Formula
- 32 How to Find the Geometric Mean (Examples)
- 33 A More Detailed Example
- 34 Why not use the Arithmetic Mean Instead?
- 35 Technology Options for Calculating the GM
- 36 Real Life Uses
- 37 Equal Ratios
- 38 A Geometric Explanation
- 39 Logarithmic Values and the Geometric Mean
- 40 Dealing with Negative Numbers
- 41 Arithmetic Mean-Geometric Mean Inequality
- 42 Antilog of a Geometric Mean
- 43 References

## Which is better geometric mean or arithmetic mean?

The arithmetic mean is always higher than the geometric mean as it is calculated as a simple average. It is applicable only to only a positive set of numbers. It can be calculated with both positive and negative sets of numbers. Geometric mean can be more useful when the dataset is logarithmic.

## What is the difference between arithmetic and geometric mean?

Arithmetic mean is defined as the average of a series of numbers whose sum is divided by the total count of the numbers in the series. Geometric mean is defined as the compounding effect of the numbers in the series in which the numbers are multiplied by taking nth root of the multiplication.

## Is arithmetic mean greater than geometric mean?

In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the

## How is geometric mean used in real life?

The growth of a bacteria increases each time and geometric mean can help us. For example, if a strain of bacteria increases its population by 20% in the first hour, 30% in the next hour and 50% in the next hour, we can find out an estimate of the mean percentage growth in population using Geometric mean.

## What type of data is the geometric mean used?

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.

## When arithmetic mean geometric mean and harmonic mean are equal?

The product of arithmetic mean and harmonic mean is equal to the square of the geometric mean. AM × HM = GM^{2}. Among the three means, the arithmetic mean is greater than the geometric mean, and the geometric mean is greater than the harmonic mean.

## Is harmonic mean smaller than geometric mean?

Harmonic mean Unless all the numbers are equal, the harmonic is always less than the geometric mean. This follows because its reciprocal is the arithmetic mean of the reciprocals of the numbers, hence is greater than the geometric mean of the reciprocals which is the reciprocal of the geometric mean.

## Is harmonic mean greater than arithmetic mean?

& (2) Harmonic mean is always lower than arithmetic mean and geometric mean. only if the values (or the numbers or the observations) whose means are to calculated are real and strictly positive.

## What are the advantages of geometric mean?

The main advantages of geometric mean are listed below: It is rigidly determined. The calculation is based on all the terms of the sequence. It is suitable for further mathematical analysis. Fluctuation in sampling will not affect the geometric mean.

## Is geometric mean affected by extreme values?

The geometric mean has an advantage over the arithmetic mean in that it is less affected by extreme values in a skewed distribution; in the above example, the arithmetic mean of the four numbers is 13, larger than the geometric mean.

## Is the geometric mean of two regression coefficient?

The coefficient of correlation is the geometric mean of the regression coefficients.

## Geometric Mean vs Arithmetic Mean

It was Lipschitz and Lerchin who discovered the functionL (x,s,a), defined by(11), and their link with Dirichlet’s famous theory on primes in arithmetic progressions was the first time they looked at this. It is instantly apparent that ForxZ,(11) simplifies to the Hurwitz Zeta function (s,a) (seeSection 2.2). Many of the features of this function may be derived from those of (z,s,a), which is a particular instance ofL (x,s,a) (z,s,a). (1) The Lerch functional equation forL (x,1s,a) may be obtained from (10) as follows: L(x,1s,a)=(s)(2s)t3rs1(|t|a;t0).

### Geometric Mean vs. Arithmetic Mean Infographics

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### Key Differences

- The arithmetic mean, also known as the additive mean, is employed in the computation of returns on a daily basis. Geometric Mean is also known as multiplicative mean, and it is a bit more hard to calculate since it requires compounding. The fundamental difference between the two means is the method by which they are computed. Calculating the arithmetic mean The arithmetic mean is the average of all of the observations in a data series, and it is defined as It is defined as the sum of all the values in a data collection divided by the total number of observations in the data set. click here to find out more This is obtained by dividing the total number of data points by the number of numbers in the dataset. When a series of numbers is determined by taking the product of these numbers and rising it to the inverse of the length of the series, this is known as the geometric mean. Return1 + Return2 + Return3 + Return4/ 4 is the formula for the geometric mean, whereas the formula for the arithmetic mean is 1
- Geometric mean (Geometric mean) It is a sort of mean that employs the product of values often ascribed to a collection of numbers to reflect the usual values or central tendency of a group of numbers. When there is an exponential change in values, this approach may be used to calculate the change. click here to find out more The geometric mean can only be computed for positive integers and is always smaller than the arithmetic mean
- On the other hand, the arithmetic mean can be calculated for both positive and negative values and is always bigger than the geometric mean The impact of outliers on a dataset is one of the most typical issues that arise when dealing with large datasets. The geometric mean in a dataset of 11, 13, 17, and 1000 is 39.5, but the arithmetic mean is 260.75 in the same dataset. The significance of the impact is clearly demonstrated. Normalization of the dataset and averaging of values are achieved as a result of the geometric mean
- As a result, no range dominates the weights and no percentage has a substantial impact on the data set. Unlike the arithmetic average, which is affected by skewed distributions, the geometric mean is not affected by them
- The arithmetic mean is employed by statisticians, but only for data sets that do not contain any notable outliers. When it comes to reading temperatures, this form of mean is really beneficial. Moreover, it is beneficial in determining the average speed of the vehicle. When a dataset is logarithmic or changes by multiples of ten, the geometric mean, on the other hand, is beneficial. Many scientists use this form of mean to characterize the size of bacteria populations since it is easy to calculate and understand. For example, the bacterial population can fluctuate between 10 and 10,000 in a single day. It is also possible to compute the distribution of income using a geometric average formula. For example, X and Y each earn $30,000 per year, but Z earns $300,000 per year. In this situation, the arithmetic average will be of no assistance. Portfolio managers draw attention to A portfolio manager is a financial market specialist who is responsible for the construction of investment portfolios on a strategic basis. a description of how and by how much a person’s wealth has risen or diminished
- Read more

### Comparative Table

Basis | Geometric Mean | Arithmetic Mean |
---|---|---|

Meaning | Geometric Mean is known as the Multiplicative Mean. | Arithmetic Mean is known as Additive Mean. |

Formula | – 1 | (Return1 + Return2 + Return3 + Return4)/ 4 |

Values | The geometric mean is always lower than the arithmetic means due to the compounding effect. | The arithmetic mean is always higher than the geometric mean as it is calculated as a simple average. |

Calculation | Suppose a dataset has the following numbers – 50, 75, 100. Geometric mean is calculated as cube root of (50 x 75 x 100) = 72.1 | Similarly, for a dataset of 50, 75, and 100, arithmetic mean is calculated as (50+75+100)/3 = 75 |

Dataset | It is applicable only to only a positive set of numbers. | It can be calculated with both positive and negative sets of numbers. |

Usefulness | Geometric mean can be more useful when the dataset is logarithmic. The difference between the two values is the length. | This method is more appropriate when calculating the mean value of the outputs of a set ofindependent eventsIndependent event refers to the set of two events in which the occurrence of one of the events doesn’t impact the occurrence of another event of the set.read more. |

Effect of Outlier | The effect of outliers on the Geometric mean is mild. Consider the dataset 11,13,17 and 1000. In this case, 1000 is the outlier. Here, the average is 39.5 | The arithmetic mean has a severe effect of outliers. In the dataset 11,13,17 and 1000, the average is 260.25 |

Uses | The geometric mean is used by biologists, economists, and also majorly byfinancial analysts. It is most appropriate for a dataset that exhibits correlation. | The arithmetic mean is used to represent average temperature as well as for car speed. |

### Conclusion

If you are dealing with percentage changes, volatile figures or data that exhibits correlation, the geometric mean is the best option. This is especially true for investment portfolios. If you make a portfolio investment instead of a single asset, you are investing in a group of assets (equity or debt), with the goal of earning returns that are proportional to the investor’s risk profile. Portfolio investments can include anything from stocks to bonds to mutual funds to derivatives to bitcoins.

The majority of financial returns, such as stock returns, bond yields, and premiums, are connected.

While arithmetic means is more suited for independent data sets because it is straightforward to use and comprehend, it is less appropriate for dependent data sets.

### Recommended Articles

This article served as a comparison of the Geometric Mean and the Arithmetic Mean. With the use of infographics and a comparison table, we’ll go through the top nine distinctions between Geometric Mean and Arithmetic Mean in this article. Check out the following articles as well if you want to learn more:

- Hopefully, this page has served as a guide to the differences between the Geometric and Arithmetic means. With the use of infographics and a comparison table, we’ll go through the top 9 distinctions between Geometric Mean and Arithmetic Mean. Check out the following articles as well if you want to learn more.

Interactions with the reader

## Geometric Mean vs Arithmetic Mean

In the realm of finance, the Arithmetic mean and the Geometric mean are the two most generally used techniques for calculating the returns on investment for investment portfolios, respectively. When people report larger returns, they do it using mathematics, which is not the most accurate way of estimating the return on investment. Because the return on investment for a portfolio over time is reliant on the returns in prior years, calculating the return on investment for a certain time period using the Geometric mean is the right method of calculating returns.

- Use of Geometric Mean vs.
- Let’s look at an example of return on investment for a $100 investment over a period of two years.
- Calculating the average return using the arithmetic mean will result in a return of 0 percent (Arithmetic mean = (-50 percent + 50 percent) /2 = 0 percent).
- A deeper look at the situation, on the other hand, paints a completely different image of the situation.
- As a result, the investor does not achieve breakeven on its investment as predicted by the arithmetic mean average, but instead suffers a loss of $25 on its investment after two years.
- Following two years, the following is the investment position: Consequently, the Geometric mean reveals the actual image of investment, which is that there has been a loss in investment with an annualized negative return of -13.40 percent compared to the historical data.
- 2.

It is possible to determine the average of a student’s marks for five topics using the arithmetic mean, because the scores of the student in different courses are independent of one another.

### Head to Head Comparison between Geometric Mean vs Arithmetic Mean (Infographics)

The following are the main eight distinctions between Geometric Mean and Arithmetic Mean:

### Key Differences between Geometric Mean vs Arithmetic Mean

Let’s have a look at some of the most significant distinctions between Geometric Mean and Arithmetic Mean:

- Neither the Geometric Mean nor the Arithmetic Mean are the techniques used to determine the returns on investment in finance, although they are both utilized in other areas such as economics and statistics. The arithmetic mean is determined by dividing the sum of the numbers by the total number of numbers in the sample size. Geometric methods, on the other hand, take into consideration the compounding impact during the calculation. In order to accurately evaluate the return on investment over a specified time period, the geometric mean should be used. Because the returns on investment for a portfolio over time are interconnected, it is important to understand how they work. The Arithmetic mean, on the other hand, is more appropriate in situations when the variables being utilized for computation are not dependent on one another. Because the arithmetic mean is more helpful and accurate when used to compute the average of a data collection with numbers that are not skewed or dependant on one another, it is more commonly utilized. Geometric means, on the other hand, are more effective and accurate in situations when there is a lot of volatility in a data set. In compared to the geometric mean, which is somewhat hard to compute and use, the arithmetic mean is comparatively simple to calculate and apply. When it comes to the realm of finance, the geometric mean is quite popular, especially when it comes to the computation of portfolio returns. However, when it comes to return calculation, an Arithmetic mean is not an effective instrument to utilize. Whenever two numbers are compared, the Arithmetic mean of the numbers is always greater than the Geometric mean of the numbers.

### Geometric Mean vs Arithmetic Mean Comparison Table

Now, let’s have a look at the top 8 comparisons between the geometric mean and the algebraic mean.

The Basis Of Comparison | Arithmetic Mean | Geometric Mean |

Definition | The arithmetic average of a series of numbers is the sum of all the numbers in the series divided by the counts of the total number in the series. | Geometric means takes into account the compounding effect during the calculation period. This is calculated by multiplying the numbers in a series and taking the nth root of the multiplication. Where n is the numbers count in series. |

Formula | If there are two numbers X and Y in the series thanArithmetic mean = (X+Y)/2 | If there are two numbers X and Y in the series thanGeometric mean = (XY)^(1/2) |

Suitability of Use | Arithmetic means shall be used in a situation wherein the variables are not dependent on each other, and data sets are not varying extremely. Such as calculating the average score of a student in all the subjects. | Geometric mean shall be used to calculate the mean where the variables are dependent on each other. Such as calculating the annualized return on investment over a period of time. |

Effect of Compounding | The arithmetic mean does not take into account the impact of compounding, and therefore, it is not best suited to calculate the portfolio returns. | The geometric mean takes into account the effect of compounding, therefore, better suited for calculating the returns. |

Accuracy | The use of Arithmetic means to provide more accurate results when the data sets are not skewed and not dependent on each other. | Where there is a lot of volatility in the data set, a geometric mean is more effective and more accurate. |

Application | The arithmetic mean is widely used in day to day simple calculations with a more uniform data set. It is used in economics and statistics very frequently. | The geometric mean is widely used in the world of finance, specifically in calculating portfolio returns. |

Ease of Use | The arithmetic mean is relatively easy to use in comparison to the Geometric mean. | The geometric mean is relatively complex to use in comparison to the Arithmetic mean. |

Mean for the same set of numbers | The arithmetic mean for two positive numbers is always higher than the Geometric mean. | The geometric mean for two positive numbers is always lower than the Arithmetic mean. |

### Conclusion

Geometric Mean and Arithmetic Mean are both used in many fields such as economics, finance, statistics, and other related fields depending on their applicability. When the variables are dependent and highly skewed, the geometric mean is a more appropriate method of finding the mean since it produces more accurate results. When the variables are not interdependent, an Arithmetic mean is used to determine the average, however when they are, an arithmetic mean is employed. As a result, in order to get the optimum effects, these two should be employed in an appropriate context.

### Recommended Articles

Economical, financial, and statistical applications for Geometric and Arithmetic Means are found in a variety of fields depending on their applicability. When the variables are dependent and highly skewed, the geometric mean is a more appropriate method of computing the mean since it produces more precise findings. Calculating the average when variables are not interdependent, on the other hand, is done using an Arithmetic mean. So in order to achieve the optimum effects, these two should be employed in the proper context.

- Geometric Mean and Arithmetic Mean are both used in economics, finance, statistics, and other fields depending on their applicability. When the variables are dependent and highly skewed, the geometric mean is more appropriate for computing the mean since it produces more accurate findings. When the variables are not interdependent, an Arithmetic mean is utilized to get the average. As a result, in order to get the optimum effects, these two should be employed in the appropriate context.

## Differences in Arithmetic & Geometric Mean

A “mean” is a mathematical phrase that refers to an average. Averages are generated in order to describe a data collection in a useful manner. Using historical data, a meteorologist may inform you that the average temperature for January 22 in Chicago is 25 degrees Fahrenheit. However, while this figure cannot forecast the precise temperature for next January 22 in Chicago, it provides enough information to know that you should bring a jacket if you plan on visiting the city on that date. The arithmetic mean and the geometric mean are two of the most widely utilized methods.

## Formulas for Calculation

In mathematics, the term “mean” refers to the average of two or more numbers or percentages. To describe a data collection in a meaningful manner, averages are generated. For example, a meteorologist may inform you that the typical temperature for January 22 in Chicago is 25 degrees Fahrenheit based on historical records. However, while this figure cannot forecast the precise temperature for next January 22 in Chicago, it provides enough information to know that you should bring a jacket if you plan on visiting the city on that day.

Understanding the distinctions between them is essential to choose which one to employ for your data.

## The Effect of Outliers

When you compare the results of arithmetic mean and geometric mean calculations, you’ll find that the influence of outliers is significantly reduced in the geometric mean calculations. What exactly does this imply? The number 1,000 is considered a “outlier” in the data set consisting of the numbers 11, 13, 17, and 1,000 since its value is much greater than all of the other numbers. It is found that the arithmetic mean is 260.25 when the result is computed. It is important to note that there is no number in the data set that is even somewhat near to 260.25, indicating that the arithmetic mean is not representative in this situation.

The influence of the outlier has been overstated in this study. The geometric mean, with a value of 39.5, performs a better job of demonstrating that the majority of the values in the data set fall between 0 and 50.

## Uses

Arithmetic means are used by statisticians to depict data that does not include any notable outliers. Because all of the temperatures in Chicago on January 22 will be between -50 and 50 degrees Fahrenheit, this form of mean is useful for depicting typical temperatures on a graph. A temperature of 10,000 degrees Fahrenheit is not likely to happen any time soon. Things like batting averages and average racing car speeds, for example, may be expressed well using arithmetic mean calculations as well.

Bacterial populations can range in size from 20 organisms one day to 20,000 organisms the next, and biologists utilize geometric methods to define their sizes.

Your income, along with that of most of your neighbors, may be roughly $65,000 per year, but what if the person up on the hill earns $65 million annually?

## Arithmetic, Geometric, and Harmonic Means for Machine Learning

The most recent update was made on August 19, 2020. A typical operation in machine learning is the calculation of the average value for a variable or a list of numbers (also known as the mean). It is an operation that you may apply on a daily basis, either directly, as when summarizing data, or indirectly, as a minor step in a bigger procedure, such as when fitting a model, as an example. The term “average” is a synonym for “mean,” which is a number that reflects the most likely value from a probability distribution in which it appears.

If you choose the wrong mean for your data, this might be a source of confusion.

Throughout this course, you will learn about the differences among the arithmetic mean, the geometric mean, and the harmonic mean.

- While the central tendency summarises the most likely value for a variable, the average is a more often used term to refer to its computation. Using the arithmetic mean is acceptable if all of the values have the same units, whereas using the geometric mean is appropriate if the values have different units. It is suitable to use the harmonic mean when the data values are ratios of two variables with distinct measurements, referred to as rates.

Start your project with my new bookStatistics for Machine Learning, which includes step-by-step explanations and the Python source codefiles for all of the sample problems. Let’s get this party started. Means for Machine Learning based on Arithmetic, Geometric, and Harmonic Functions Some rights retained for this photo taken by Ray in Manila.

## Tutorial Overview

This lesson is broken into five sections, which are as follows:

- What is the average
- What is the arithmetic mean
- What is the geometric mean
- What is the harmonic mean
- How do you choose the correct mean

## What Is the Average?

Central tendency is a single number that reflects the most frequent value for a set of values. It is represented by the letter C. In more technical terms, it is the value that has the highest likelihood of occurring from the probability distribution that specifies all potential values that a variable might have. When analyzing a data sample, there are numerous ways to calculate the central tendency, including themean, which is calculated from the values, themode, which is the most common value in the data distribution, and themedian, which is the middle value if all the values in the data sample were ordered.

They can both be used in the same sentence.

There are a variety of approaches to compute the mean depending on the type of data being considered.

There are more methods, as well as many more central tendency measures, but these three methods are likely the most widely used in the field of statistics (e.g. the so-calledPythagorean means). Now, let’s take a closer look at each of the mean computations one at a time.

## Arithmetic Mean

The arithmetic mean is computed by dividing the sum of the values by the total number of values, denoted by the letter N. Another method of calculating the arithmetic mean is to compute the total of the values and multiply that sum by the reciprocal of the number of values (1 over N); for example: For data samples where all values have the same units of measure, such as heights, dollars, kilometers or other units of measurement, the arithmetic mean is used to calculate the average. When computing the arithmetic mean, the numbers can be either positive or negative, or they can both be positive and negative.

multiple peaks, a so-called multi-modal probability distribution).

When a variable has a Gaussian or Gaussian-like data distribution, this is more significant than when the variable does not.

How to calculate the arithmetic mean for a list of 10 numbers is demonstrated in the following example:

example of calculating the arithmetic meanfromnumpyimportmeandefine the datasetdata=calculate the meanresult=mean(data)print(‘Arithmetic Mean: %.3f’%result) |

When the example is run, the arithmetic mean is calculated and the result is reported.

## Geometric Mean

Calculated as the N-th root of the product of all values, where N is the total number of values, the geometric mean is defined as For example, if the data set has just two values, the geometric mean is defined as the square root of the product of the two values. The cube-root is used for the first three numbers, and so on. When the data comprises values in several units of measure, such as height, money, miles, and so on, the geometric mean is the most acceptable choice to make. The geometric mean does not allow negative or zero values, thus all numbers must be positive, as seen in the example above.

The G-Mean(geometric mean) metric is an evaluation metric that is calculated as the geometric mean of the sensitivity and specificity metrics.

Example 1 shows how to find the geometric mean of a list of ten numbers using the formula shown below.

example of calculating the geometric meanfromscipy.statsimportgmeandefine the datasetdata=calculate the meanresult=gmean(data)print(‘Geometric Mean: %.3f’%result) |

When the example is run, the geometric mean is calculated and the result is reported.

## Harmonic Mean

When the example is run, the geometric mean is calculated and the result is reported. A simpler computation of the harmonic mean may be performed when there are just two values (x1 and x2), and the result is as follows: A simpler computation of the harmonic mean may be performed when just two values (x1 and x2) are available:

example of calculating the harmonic meanfromscipy.statsimporthmeandefine the datasetdata=calculate the meanresult=hmean(data)print(‘Harmonic Mean: %.3f’%result) |

When the example is run, the harmonic mean is calculated and the result is reported.

## How to Choose the Correct Mean?

There are three distinct approaches to calculating the average or mean of a variable or dataset that we have discussed. Arithmetic mean is the most widely used mean, yet it may not be applicable in all situations. Each mean is suited for a certain sort of data; for example, the following:

- The arithmetic mean should be used when values have the same units as each other. If the values have different units, the geometric mean should be used. If the values are rates, the harmonic mean should be used.

The geometric and harmonic means cannot be employed directly if the data contains negative or zero values, which is the case in most cases.

## Further Reading

If you want to learn more about a certain issue, you may find more resources in this area.

### APIs

- You may learn more about this topic by reading the materials provided in this area.

### Articles

- Average, according to Wikipedia
- Central tendency, according to Wikipedia
- Arithmetic mean, according to Wikipedia
- Geometric mean, according to Wikipedia
- Harmonic mean, according to Wikipedia

## Summary

This course taught you how to distinguish between the arithmetic mean, the geometric mean, and the harmonic mean, among other concepts. You gained knowledge in the following areas:

- While the central tendency summarises the most likely value for a variable, the average is a more often used term to refer to its computation. Using the arithmetic mean is acceptable if all of the values have the same units, whereas using the geometric mean is appropriate if the values have different units. It is suitable to use the harmonic mean when the data values are ratios of two variables with distinct measurements, referred to as rates.

Do you have any queries for us right now? Ask your questions in the comments section below, and I’ll try my best to respond as soon as possible.

## Get a Handle on Statistics for Machine Learning!

.by writing lines of code in the Python programming language Learn how in my new Ebook, Statistical Methods for Machine Learning, which is available now. Self-study lessons on topics like as Hypothesis Testing, Correlation, Nonparametric Statistics, Resampling, and many more are available on the site.

#### Discover how to Transform Data into Knowledge

Leave the academics to the professionals. There are just results. Take a look at what’s within.

## Question Corner – Applications of the Geometric Mean

Panel de navigation: (These buttons explainedbelow) On May 22, 1997, Senthil Manick posed the following question: When would it be appropriate to utilize the geometric mean rather than the arithmetic mean? In general, what is the function of the geometric mean and why is it important? When numerous values are added together to generate a total, the arithmetic mean is important to consider. Arithmetic mean provides a solution to the question, “If all of the variables were equal, what would the value of the mean be in order to obtain the same total?” Similar to this, the geometric mean is useful whenever a number of variables multiply to yield a single product.

- What is the typical rate of return on this investment?
- The geometric mean of these three integers is the amount that is significant.
- If you compute the geometric mean, you will obtain roughly 1.283, which means that the average rate of return is approximately 28% on the investment (not 30 percent which is what the arithmetic mean of 10 percent , 60 percent , and 20 percent would give you).
- Listed below are some fundamental mathematical truths regarding the arithmetic and geometric means: Consider the following scenario: we have two quantities, A and B.
- According to one interpretation (which is probably the most widely accepted), this quantity represents the halfway between the two integers when regarded as points on a line.
- A second way to think about the arithmetic mean is to think of it as the length of the sides of a square whose perimeter is the same as our rectangle.
- In mathematics, it is well known that the geometric mean is always smaller than or equal to the arithmetic mean (with the equivalence occurring only when A = B).

It should be noted that this inequality may be rather powerful and that it does appear in the proofs of several calculus results from time to time.

Ellis, a student at Southeast Bulloch High School: Could you please provide the formula for the geometric mean of a series of numbers if I’m trying to calculate the compound annual growth rate for a series of numbers that includes negative figures?

As a result, the geometric mean of numbers is equal to the square root of the product.

For example, if you’re looking at an investment that rises by 10% one year and declines by 20% the next, the simple rates of change are 10% and -20%, but that’s not what you’re looking at when you’re considering the geometric mean of the investment.

At the conclusion of the second year, you have 0.8 times the amount of money you had at the beginning of the second year (the original minus one fifth of it).

This means that the mean is around 0.938.

This results in (roughly) -6.2 percent compound annual growth rate.

I have discovered that the terms “average percentage growth” and “average % growth rate” have drastically different definitions in the research, and in many instances, I am unable to discern between the usage of the words “growth” and “growth rate” in the literature.

However, while I, along with the majority of those I spoke with, first thought that these GICs were ensuring a rate of return that was equivalent to the rate of growth of the index, the bank had a different meaning.

Rather than considering the growth by month-end for each month (which is 10 for the first month, 20 for the second, and so on, with this growth reaching 120 by the twelfth month), the bank appears to be taking an average of these growths, resulting in a “average growth” of 60 and an average percentage growth of 60/1000 or 6 percent.

In terms of other definitions, there appears to be widespread agreement on the following:

- It was a genuine growth rate of 120/1000 or 12 percent. The average percentage growth rate every year is 12 percent
- However, the rate might vary. A 12-percent compound annual growth rate is seen
- Nonetheless,

It appears to me that averaging growth, as done by the bank, is inappropriate in this case. Because an average should be comprised of equal periods, in the bank’s calculation, we average growth for the first month with growth for the first 11 months to arrive at an average growth for the first month. Because the index units increased by 10 each month, should the average growth rate not be 10 units each month as well? If so, how is this different from the average growth rate of 10 units per month?

- When it comes to growth and growth rate, what, if any, is the right difference to make?
- First, let’s try to get our heads around the terms.
- The term “growth” is frequently used in a broad sense to refer to any of the notions listed above.
- In your example, the growth for the period comprising of the first month was ten, and the growth for the period consisting of the first twelve months was one hundred and twenty percent.
- It is possible that it will change over time.
- If the growth rate remains constant throughout a period of time, then the average growth rate for that period will be the same as the constant number from the beginning of the period.
- The average monthly growth rate throughout all time periods was similarly 10 units per month on average.

During the whole year, the average growth rate was 120 units every 12 months, which equates to 10 units each month.

After instance, if a ten-dollar investment increased in value by one dollar over a year (an average growth rate of one dollar per year), you would anticipate a ten-thousand-dollar investment to increase in value by one thousand dollars per year: a vastly different annual growth rate!

In a period of time, the percentage growth is calculated as a ratio of the growth to the initial value.

During the second month, the percentage increase rate was around 0.99 percent (10/1010).

When comparing a percentage growth rate to a current value, we have the term “percentage growth rate.” If the index value is 1000 and it is rising at a pace of 10 units each month, the index is experiencing a percentage growth rate of 1 percent per month, as seen in the example above.

And so forth.

The only problematic part about percentage growth rate is that it varies depending on the unit of measurement: a percentage growth rate of 1 percent per month is not the same as a percentage growth rate of 12 percent per year.

A percentage growth rate of one percent per month is the same as a percentage growth rate of around 12.7 percent per year, as seen in the chart below.

If your units are years, then T =1.G= 12 percent, which means that the average percentage growth rate isper year.

If your units are months, the average percentage growth rate is 0.949 percent each month, which is approximately 0.949 percent annually.

When people in the investing community talk about “growth” or “growth rate,” they are frequently referring to percentage growth and percentage growth rate, respectively.

As you point out, averaging the growth rates over time periods of widely varying lengths is not especially helpful in terms of inferring trends.

What is appropriate, however, is to average the growth rates over overlapping time periods of equal durations, in order to smooth out changes in the index’s value over time.

You wouldn’t want to be able to claim that there was no development at all throughout the course of the year just because the day you chose for the evaluation of the index happened to be a lousy one, would you?

As a result, it is usual practice to take an average of the index value over a certain period of time.

This is because the index’s average1996 value is the same as the index’s average1997 value.

Now, if the index remained at 1000 for the whole year 1996 and then increased to 1120 in the manner you describe during 1997, you would only be receiving a 6 percent return from the average 1996 value (1000) to the average 1997 value (1120), which is a disappointing result (1060).

Another option that could be appropriate is to take an average of the percentage increases across several 12-month periods that all finish in the same year.

Due to the fact that the percentage growth rates are being averaged across equal but overlapping time periods, this is an appropriate type of average to use.

Suppose the index began 1996 at 1000 and increased consistently at a rate of 10 units per month, reaching almost 1240 by the end of 1997, before collapsing to 1150 on the final day of 1997 owing to a sudden correction in the markets.

Despite the fact that the increase from December 31, 1996 to December 31, 1997 was just 2.7 percent on an annualized basis (from 1120 to 1150), the growth from November 30, 1996 to November 30, 1997 was about 11 percent (from 1110 to 1230).

According to my suspicions, the GIC you are worried about is most likely using some form of acceptable averaging, such as the one described above, and that the bank has miscommunicated the method of computation to you.

In other words, you will experience the unreasonably high returns during the first year of the investment, but these effects will be negligible over the long term.

Certainly, this represents average development across periods of varying lengths, but the periods range from 9 years to 10 years, which is a far smaller variation than your example, in which the periods span from 1 month to 12 months!

However, I hope that this has helped to clarify some of the mathematical concepts involved.

Philip Spencer is the original web site creator and developer of mathematical content.

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## Arithmetic vs Geometric Mean: Which to use in Performance Appraisal

The mean return is used in the computation of the majority of performance assessment metrics. This can take the shape of a geometric mean or a straightforward arithmetic average. The fact that both sorts of measures are capable of being employed raises the question of which measure should be implemented. When it comes to calculating performance, we are used to calculating returns in a geometrical manner (i.e., including compounding). As a result, many investment managers prefer to utilize the geometric mean in appraisal calculations since it is more convenient to use the stated time-weighted return rather than calculating the arithmetic mean individually.

- It is recommended that when computing performance assessment measures that relate return to risk, such as Sharpe ratio, the return utilized in the numerator of the ratio should be the arithmetic mean of the return stream rather than the geometric mean of the return stream.
- Let’s have a look at a straightforward example that exhibits this phenomenon: In contrast to arithmetic mean, the geometric mean of strategies with considerable volatility is lower (7.5 percent vs.
- This is due to the fact that the geometric mean penalizes the return stream for taking risks.
- 14.2 percent for Portfolio 2).
- The inclusion of risk in the numerator is unnecessary since it has already been accounted for in the denominator; in fact, incorporating it would be equivalent to double-counting the risk that has already been undertaken.
- When comparing risk-scaled performance metrics, utilizing the geometric return will not always have a significant impact on the conclusion, but using the arithmetic mean is more accurate in terms of technical accuracy.

## Geometric Mean: Definition, Examples, Formula, Uses

Statistics Terms and Definitions Geometric Mean (Geometric Mean)

### Contents.

- How to Calculate the Geometric Mean
- Technology Options
- Applications in Real Life

More In-Depth Information:

- Equivalence ratios and a geometric explanation
- Logarithmic values and dealing with negative numbers
- Equivalence ratios and a geometric explanation Inequality between the Arithmetic Mean and the Geometric Mean
- The antilog of a Geometric Mean is defined as

## Geometric Mean Definition and Formula

For three examples of how to find the geometric mean, please see the video below. Are you unable to view the video? To learn more, please visit this page. The geometric mean is a sort of average that is commonly used to represent growth rates, such as population growth or interest rates, among other things. Instead of adding sitesites, the geometric means multiplying sitesites. Furthermore, the geometric mean can only be obtained for positive values. As is true of most things in mathematics, there is an easy explanation and a more, ahem, mathematically correct method of expressing the same thing.

The apostrophe in the formula represents product notation: “Product” in mathematics is denoted by the mathematical notation, which is related to the (perhaps more common) “summation notation.”

## How to Find the Geometric Mean (Examples)

Do you require assistance with a homework question? Check out our tutoring website for more information! The following is an example of how to use the term “example.” Is there a geometric mean for the numbers 2, 3, and 6? The cubed root (since there are three numbers) is obtained by multiplying the integers together and then taking the square root of the result = (2*3*6) 3.30 divided by third It is important to note that the power of (1/3) is the same as the cubed root3. Change the denominator in the fraction to whichever “n” you have on hand to convert an nth root to this notation.

- The 5th root is equal to the (1/5) power, the 12th root is equal to the (1/12) power, and the 99th root is equal to the (1/99) power.

Example 2: What is the geometric mean of the numbers 4,8,3,9, and 17 in this equation? Take the 5th root (since there are 5 numbers) by multiplying the numbers together: (4*8*3*9*17) (1/5) = 6.81 (because there are 5 numbers). Example 3: What is the geometric mean of the fractions 1/2, 1/4, 1/5, 9/72, and 7/4? a To begin, add the numbers together and then calculate the 5th root: (1/2*1/4*1/5*9/72*7/4) (1/5)= 0.35 (1/2*1/4*1/5*9/72*7/4) (1/5)= 0.35 Example 4: Over the course of 10 years, the typical person’s monthly wage in a certain municipality increased from $2,500 to $5,000.

Solution: Step 1: Calculate the geometric mean of the data.

Step 2: Multiply by ten (to get the average increase over ten years).

According to the General Motors, the average increase is 353.53.

## A More Detailed Example

Consider the following scenario: you possess a piece of art that grows in value by 50% the first year after you purchase it, 20% the second year, and 90% the third year. What these numbers tell you is that the value at the end of the first year has been multiplied by 150 percent, or 1.5, the value at the end of the second year has been multiplied by 120 percent, or 1.2, and the value at the end of the third year has been multiplied by 190 percent, or 1.9, at the end of the third year. The geometric mean, which can be computed as (1.5*1.2*1.9) (1/3)= 1.50663725458.

In other words, if you doubled your initial investment by 1.51 each year, you would end up with the same amount as if you multiplied it by 1.5, 1.2, and 1.9 at various points along the way.

- Consider the following scenario: you possess a work of art that grows in value by 50% the first year after you purchase it, 20% the second year, and 90% the third year. What these numbers tell you is that the value at the end of the first year has been multiplied by 150 percent, or 1.5, the value at the end of the second year has been multiplied by 120 percent, or 1.2, and the value at the end of the third year has been multiplied by 190 percent, or 1.9, since the beginning of the first year. The geometric mean, which can be computed as (1.5*1.2*1.9) (1/3)= 1.50663725458. or around 1.51, is what you are aiming for when the numbers are multiplied. In other words, if you doubled your original investment by 1.51 each year, you would end up with the same amount as if you multiplied it by 1.5, 1.2, and 1.9 at different intervals throughout time.

Alternatively, use the geometric mean: $90,000 multiplied by 1.50663725458 is $307,000** If you perform this calculation, the result will be somewhat different according to the amount of decimal places I specified above. In other words, using a calculator, you should be able to obtain the identical answer. Return to the top of the page

## Why not use the Arithmetic Mean Instead?

(1) The arithmetic mean is the product of the total of the data items divided by the number of data items in the set: (1.5+1.2+1.9)/3 = 1.53 The geometric mean is the sum of data items divided by the number of data items in the set: In this case, as you can undoubtedly guess, adding 1.53 to your starting price will not get you anywhere, and multiplying it will give you the incorrect answer.

$90,000 multiplied by 1.53 multiplied by 1.53 multiplied by 1.53 multiplied by 1.53 Equals $322,343.91

## Technology Options for Calculating the GM

- Although JMP does not come with a formula, you may write one using theformula editor. Use the ALLGEO statistic keyword in the PROC SURVEYMEANS statement if you’re using SAS/STAT 12.1 or later to do your analysis. Versions prior to this one do not have this functionality
- In Excel, you can use theGEOMEAN function to get the mean of any positive data range. The syntax is as follows: GEOMEAN(number1,.)
- If you’re using SPSS, you should use theMEANScommand. To perform an analysis, go to the SPSS menus and pick Analyze Compare Means Means, then click on the “Options” button and choose from the list of available statistics on the left
- MINITAB: The GMEAN function should be used. This is represented by the syntax GMEAN(number), where “number” is the column number. All numbers must be in the positive direction. MAPLE: The calling sequence GeometricMean provides a number of different choices for computing the mean from a data sample (A), a matrix data set (M), or a random variable or distribution (R) (X). The whole set of settings may be found in this article. There is no built-in function for the TI83. You may work around this problem by entering your information into a list, and then entering the geometric mean formula on the home screen. The TI89 is similar to the TI83 in that it does not have a built-in function. For example, you could get an app such as the ” Statistics and Probability Made Easy ” app, which I personally endorse because I used it while in graduate school:)

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## Real Life Uses

Choosing aspect ratios in film and video has always relied on the geometric mean (the proportion of the width to the height of a screen or image). It is used to find a compromise between two aspect ratios by distorting or cropping both ratios in the same proportion.

### Computer Science

A staggering quantity of data is generated by computers, much of which must be summarized using statistical methods. One study examined the precision of different statistics (arithmetic means, geometric means, and percentage in the top x percent) for a mind-boggling 97 trillion bits of citation data, which was a mind-boggling amount of information. According to the findings of the investigation, the geometric mean was the most exact (seethis Cornell University Library article).

### Geometry

1. Proportional to the Mean In geometry, the geometric mean is utilized as a percentage (it is often referred to as the “mean proportional” or “mean proportional”). The positive number x is the mean proportionate of two positive numbers a and b, resulting in the following: When attempting to solve this percentage, the formula x=a*b is used. Triangles ADC, ADB, and CAB are similar in appearance in this illustration. The proportion may be used to discover missing sides of triangles that are similar to each other.

Secondly, the Golden Ratio The geometric mean and other comparable rectangles may be used to calculate the golden mean, which has a value of around 1.618.

These two rectangles are both composed of the golden ratio, which is the relationship between the rectangle’s length and breadth.

### Medicine

In medicine, the Geometric Mean has a wide range of uses. When it comes to certain metrics, such as the computation of gastric emptying timesJNM, it has been referred to as the “gold standard.”

### Proportional Growth

A number of applications for the Geometric Mean may be found in the medical field. When it comes to some metrics, such as the computation of stomach emptying periods, it has been referred to as the “gold standard.”JNM

### United Nations Human Development Index

The Human Development Index (HDI) is a measure of a country’s growth that takes into consideration variables other than economic development when calculating its progress. It is defined as “.a summary assessment of average performance in major areas of human development, such as living a long and healthy life, being knowledgeable, and having a reasonable quality of living.”. For each of the three variables, the HDI is the geometric mean of the normalized indices. UNDP It is referred to as “normalized” indexes, and it refers to the fact that the geometric mean is not influenced by variations in scoring indices.

The geometric mean is unaffected by any of these variables.

### Water Quality Standards

The geometric mean of test findings for water quality (particularly, fecal coliform bacteria concentrations) is occasionally used to represent the results. Authorities in charge of water management choose a geometric mean over which beaches or shellfish beds must be closed. In water quality estimates, the damping effect of the geometric mean is particularly important, according to CA.GOV, because bacteria levels can vary by a factor of ten to one thousand thousand over a period of time. Return to the top of the page

## Equal Ratios

There is an intriguing phenomenon that occurs when you examine the geometric mean and the values you entered into the computation. Consider the following scenario: you wish to calculate the geometric mean of the numbers 4 and 9. To do the math, multiply 4 by 9 and get the answer of 6.

- (4) The ratio of the first number (4) to the geometric mean (6) is 4/6, which may be reduced to 2/3. Sixteen to six is the ratio of the second number (9) to the geometric mean (6), which decreases to two-thirds.

Notice that the ratios are the same as they were before. You can see from this that the geometric mean is a type of “average” of all of the multipliers that you are including into your equation. Take, for example, the integers 2 and 18. The center number should be something like this: what number could you put there such that the ratio of 2 (to this number) is the same as the ratio of this number to 18. 2 (or more?) 18 If you picked 6, you are correct, since 2 * 3 = 6 and 6 * 3 = 18 are the same number.

It is possible to obtain the same answer (6) by applying the formula, thus if you are ever confronted with a problem similar to the one above in a math class, all you need to do is determine the square root of the integers multiplied together: ((2 * 18)) = 6 is the answer.

## A Geometric Explanation

As you can see, the ratios are the same for both groups. You can see from this that the geometric mean is a type of “average” of all of the multipliers that you are including into the equation. To illustrate, consider the integers 2 and 18. Is there a number that you could put in the middle of the circle such that the ratio of 2 (to this number) is the same as the ratio of this number to 18? the number two? 18 Your prediction of 6 is correct since 2 * 3 = 6 and 6 * 3 = 18 and 2 * 3 = 6 is the same as 18.

It is possible to obtain the same answer (6) by applying the formula, therefore if you are ever confronted with a problem similar to the one above in a math class, all you need to do is calculate the square root of the integers multiplied together.

## Logarithmic Values and the Geometric Mean

If you think about it, the geometric mean is just the average of logarithmic numbers that have been transformed back to the base 10. Especially if you’re already familiar with logarithms, this might be a really straightforward method to approach the problem.

Consider the following scenario: you wish to get the geometric mean of the numbers 2 and 32. The first step is to create a plan. Using base 2 logs (you may potentially use any base), convert the following integers to base 2 logs:

Step 2: Calculate the (arithmetic) mean of the exponents obtained in Step 1. The average of 1 and 5 is equal to three. Due to the fact that we are still working in base 2, our average is 2 3, which gives us the geometric mean of 2 * 2 * 2 = 8 as a result. Return to the top of the page

## Dealing with Negative Numbers

In most cases, only positive values may be used to calculate the geometric mean. If you have negative numbers (which is common in the investing world), it is feasible to obtain a geometric mean, but you must first perform some preliminary arithmetic (which is not always straightforward!). Example: What is the geometric mean of an investment that exhibits a 10 percent increase in year one and a 15 percent drop the following year? Step 1: Calculate the total amount of growth that the investment will see over the course of each year.

The next year, you have 90 percent (or 0.9) of the money you had at the beginning of year 2.

In this case, GM = (1.1 * 0.9) = 0.99.

Every year, at a rate of around 1 percent, your investment is gradually losing money.

## Arithmetic Mean-Geometric Mean Inequality

The Arithmetic Mean-Geometric Mean Inequality (AM-GM inequality) says that, for a list of non-negativereal numbers, the arithmetic mean is greater than or equal to the geometric mean in a given direction. Using the mean formulae for both forms of mean, we can get the following inequality: Consider the following series of numbers: the arithmetic mean is 25, which is bigger than the geometric mean, which is 18. The several proofs of this inequality are beyond the scope of this statistics website, however Bjorn Poonenprovides this straightforward one-line demonstration for the AM-GM inequality with two variables: Return to the top of the page

## Antilog of a Geometric Mean

The antilog of a number is the number raised by a factor of ten. Consider the case where your geometric mean is 8. Increase the basic time from 8:10 to 8:10. The general formula is as follows: antilog(g) = 10 g= 10 antilog(a*b) = 10 g= 10 a*b Return to the top of the page

## References

An Introduction to Categorical Data Analysis, A. Agresti, 1990. The publisher is John Wiley and Sons in New York. Klein, G., et al (2013). The Statistical Cartoon as an Introduction. HillWamg. S. Kotz and colleagues (2006) published the Encyclopedia of Statistical Sciences by Wiley. W.P. Vogt & Sons, Inc. (2005). The Statistics Dictionary is a collection of statistical terms and definitions. An Introduction to Methodology for the Social Sciences, written in a nontechnical style. SAGE.- Do you require assistance with a homework or exam question?

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