## Geometric Mean vs Arithmetic Mean Differences

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 whereas Arithmetic Mean is simply the average and is calculated by adding all the numbers and divided by the count of that series of numbers. In this article, we look at the key differences between Arithmetic vs Geometric mean.

### Geometric Mean vs Arithmetic Mean Infographics

Here we provide you with the top 9 difference between Geometric Mean vs Arithmetic Mean

### Geometric Mean vs Arithmetic Mean – Key Differences

The key differences between Geometric Mean vs Arithmetic Mean are as follows –

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- 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 are little complicated and involves compounding
- The main difference in both these means is the way it is calculated. The arithmetic mean is nothing but an average. It 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 is 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 have a significant effect on the data set. The geometric mean is not influenced by skewed distributions as the arithmetic average is.
- Consider the returns for Asset X shown over a 5-year time horizon. We will use this data to understand the difference in calculating arithmetic and geometric mean. Using the formula above, the arithmetic

Mean is calculated as

**Arithmetic mean = [20% + 15% + 7% + 8% + 12%]/5 = 46%/10 = 14.24%**

**Geometric Mean =10√(1+20%)(1+15%)(1+7%)(1+8%)(1+12%) = 11%**

- 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 useful 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 yearly. In this case, the arithmetic average will not be useful. Portfolio managers also use geometric mean to highlight how the wealth and by how much wealth of an individual has increased or decreased.

### Geometric Mean vs Arithmetic Mean Head to Head Difference

Let’s now look at the head to head difference between Geometric Mean vs Arithmetic Mean

Basis |
Geometric Mean |
Arithmetic Mean |
||

Meaning |
Geometric Mean is known as 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 mean due to the compounding effect | The arithmetic mean is always higher than 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 |
The geometric mean is applicable only to an only positive set of numbers | The arithmetic mean can be calculated with both positive and negative set 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, economist and also majorly by financial analysts. It is most appropriate for a dataset that exhibit correlation | The arithmetic mean is used to represent average temperature as well as for car speed |

### Conclusion

Use of geometric mean is appropriate for percentage changes, volatile numbers and for 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 important 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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