Arithmetic Mean vs Geometric Mean with Formula

In the realm of finance, the Arithmetic mean and Geometric mean are commonly employed to compute the returns on investment for investment portfolios. People claim larger returns using the arithmetic mean, which is not the correct way to calculate the return on investment. The Geometric Mean is the correct approach to compute the return on investment for a certain time period because the return on investment for a portfolio over time is dependent on returns in previous years. The arithmetic mean is better suited to situations when the variables used to calculate the average are not dependent on one another.

What is Arithmetic Mean Vs Geometric Mean

The list below shows the formulas for the arithmetic and geometric means. For example, suppose you have a set of data values, x₁, x₂, x₃, ….. x_n,

• The arithmetic mean (AM) = (x₁ + x₂ + x₃ + … + x_n) / n.

• The geometric mean (GM) = (x₁ · x₂ · x₃ · … · x_n)^1/n.

Example: For the values 1, 3, 5, 7, and 9:

Arithmetic mean = (1 + 3 + 5 + 7 + 9) / 5 = 5.

Geometric mean = (1 × 3 × 5 × 7 × 9)^1/5 ≈ 3.93.

As a result, the arithmetic mean equals the total number of values divided by the sum of the values. To put it another way, the arithmetic mean is simply the average of the values. The geometric mean, on the other hand, is the product of the values multiplicatively inversed by the total number of values. In terms of meaning and formula, this is the distinction between AM and GM. There are numerous discrepancies between arithmetic and geometric means.

The difference in terms of results

The geometric mean is always less than (or equal to) the arithmetic mean for a set of data values. As we can see in the case above, 3.93 (GM) < 5 (AM).

The difference in terms of data values

The geometric mean is only valid for positive numbers, but the arithmetic mean is valid for both positive and negative numbers.

The difference in terms of the effect of outliers

The outlier has little effect on the geometric mean, but on the arithmetic mean it has a greater effect. Consider a group of data values that includes an outlier, such as 10, 12, 14, and 99. Let’s figure out AM and GM.

AM = (10 + 12 + 14 + 99) / 4 = 33.75

GM = (10 × 12 × 14 × 99)^1/4 ≈ 20.19.

We can see that the majority of the data values are significantly different from AM, although GM is unaffected.

The difference in terms of Ease of Use

The geometric mean is difficult to use because it requires the product and taking roots, whereas the arithmetic mean is simple to use because it involves the sum.

The difference in terms of Accuracy

When the data values are not skewed and independent of one another, AM is accurate. When there is a lot of variation in the data, GM is more accurate.

The difference in terms of Application

Statistics, economics, history, and sociology all use the arithmetic mean (which is nothing more than average). In finance, the geometric mean (which is just compounded growth) is used to compute average growth rates.

CONCLUSION

Arithmetic mean

The arithmetic mean is the average of a series of numbers whose sum is divided by the total count of the numbers in the series. (x + y)/2 is the formula. This refined application in daily calculations with a consistent set of facts.

Geometric mean

The compounding effect of the numbers in a series multiplied by taking the nth root of the multiplication is defined as the geometric mean. It’s written as (xy)^(1/2). This refined application in financial portfolio returns calculation.