What is the Difference Between Geometric Mean and Arithmetic Mean?

The main difference between geometric mean and arithmetic mean lies in the way they are calculated and their applications. Here are the key differences:

1. Calculation:

• Arithmetic Mean: Calculated by adding all the values and dividing by the total number of values. The formula is: $$\text{Arithmetic Mean} = \frac{x1 + x2 + \dots + xn}{n}$$, where $$xi$$ are the values and $$n$$ is the number of values.
• Geometric Mean: Calculated by multiplying all the values and raising their product to the power of $$\frac{1}{n}$$. The formula is: $$\text{Geometric Mean} = \left(x1 \times x2 \times \dots \times xn\right)^{\frac{1}{n}}$$, where $$xi$$ are the values and $$n$$ is the number of values.

1. Applications:

• Arithmetic Mean: Widely used in fields like statistics, economics, history, and sociology. It is a commonly used measure of central tendency and is often referred to simply as "the mean".
• Geometric Mean: Used to calculate the average growth or average percentage change, particularly in financial indices and population growth rates. It is considered more accurate for percentage change and positively skewed data.

1. Effect of Compounding:

• Arithmetic Mean: Does not take into account the effect of compounding. It simply adds and divides the values.
• Geometric Mean: Takes into account the effect of compounding by multiplying the values and raising their product to the power of $$\frac{1}{n}$$.

1. Positive and Negative Values:

• Arithmetic Mean: Applies to both positive and negative values.
• Geometric Mean: Applies only to positive values. It does not accept negative or zero values.

1. Accuracy:

• Arithmetic Mean: Accurate when the data values are not skewed and are independent of each other.
• Geometric Mean: More accurate when there is volatility in the data.

In summary, the arithmetic mean is a simple average of the values, adding and dividing them, while the geometric mean is calculated by multiplying the values and taking their root. The geometric mean is more appropriate when dealing with growth rates and positively skewed data, whereas the arithmetic mean is more suitable for data with no skew and no compounding effect.

Comparative Table: Geometric Mean vs Arithmetic Mean

The main differences between the geometric mean and the arithmetic mean are the methods of calculation and the type of data they are suitable for. Here is a table comparing the two:

In summary, the geometric mean is suitable for calculating the average growth rate or returns when there are different units of measure or positively skewed data, while the arithmetic mean is appropriate for calculations when the items in the dataset are the same unit type and there is no volatility in the data.