Geometric Mean vs Arithmetic Mean

Differences Between Geometric and Arithmetic Mean

Geometric mean is the calculation of mean or average of series of values of product which takes into account the effect of compounding and it is used for determining the performance of investment whereas arithmetic mean is the calculation of mean by sum of total of values divided by number of values.

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The geometric mean is calculated for a series of numbers by taking the product of these numbers and raising it to the inverse length of the series. Arithmetic Mean is simply the average and is calculated by adding all the numbers and divided by the count of that series of numbers.

Geometric Mean vs. Arithmetic Mean Infographics

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For eg:
Source: Geometric Mean vs Arithmetic Mean (wallstreetmojo.com)

Key Differences

  • The arithmetic mean is known as additive mean and are used in the everyday calculation of returns. Geometric Mean is known as multiplicative mean and is a little complicated and involves compounding.
  • The main difference in both these means is the way it is calculated. The arithmetic meanArithmetic MeanArithmetic mean denotes the average of all the observations of a data series. It is the aggregate of all the values in a data set divided by the total count of the observations.read more is calculated as the sum of all the numbers divided by the number of the dataset. The geometric mean is a series of numbers calculated by taking the product of these numbers and raising it to the inverse of the length of the series.
  • Formula for geometric mean is {[(1+Return1) x (1+Return2) x (1+Return3)…)]^(1/n)]} – 1 and for arithmetic mean is (Return1 + Return2 + Return3 + Return4)/ 4.
  • Geometric mean can only be calculated for positive numbers and is always less than geometric meanwhile, arithmetic mean can be calculated for both positive and negative numbers and is always greater than the geometric mean.
  • A most common problem with having a dataset is the effect of outliers. In a dataset of 11, 13, 17, and 1000 the geometric mean is 39.5, while the arithmetic means 260.75. The effect is clearly highlighted. Geometric mean normalizes the dataset, and the values are averaged out; hence, no range dominates the weights, and any percentage does not significantly affect the data set. The geometric mean is not influenced by skewed distributions as the arithmetic average is.
  • The arithmetic mean is used by statisticians but for data set with no significant outliers. This type of mean is useful for reading temperatures. It is also helpful in determining the average speed of the car. On the other hand, the geometric mean is useful in cases where the dataset is logarithmic or varies by multiples of 10.
  •  Many biologists use this type of mean to describe the size of the bacterial population. For example, the bacterial population can be 10 in one day and 10,000 on others. Income distribution can also be calculated using a geometric average. For example, X and Y make $30,000 yearly, while Z makes $300,000 annually. In this case, the arithmetic average will not be useful. Portfolio managers highlightPortfolio Managers HighlightA Portfolio Manager is an executive responsible for making investment decisions & handle investment portfolios for fulfilling the client’s investment-related objectives. Also, he/she works towards maximizing the benefits & minimizing the potential risks for clients. read more how the wealth and by how much wealth of an individual has increased or decreased.

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) x (1+Return2) x (1+Return3)…)]^(1/n)]} – 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 of independent events.
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 by financial 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

The use of geometric mean is appropriate for percentage changes, volatile numbers, and data that exhibit correlation, especially for investment portfolios. Most returns in finance are correlated like stocks, the yield on bonds, and premiums. The longer period makes the effect of compounding more critical and hence also the use of a geometric mean. While for independent data sets, arithmetic means is more appropriate as it is simple to use and easy to understand.

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