Geometric Mean

Introduction to Geometric Mean

The geometric mean (GM) is a way of calculating an average, but instead of adding values like the regular (arithmetic) mean, it multiplies them and then takes a root. The geometric mean is defined as the $n$th root of the product of $n$ positive values.

If we have two observations let’s say 9 and 4, then the geometric mean is the square root of the product of these values, which is 6 ($\sqrt{9\times 4}=6$. If there are three values let’s say  3, 9, and 3 then the geometric average will be the $sqrt[3]{3\times 9 \times 3} = 3$. In a similar pattern, mathematically, for $n$ number of observations ($x_1, x_2, \cdots, x_n$) then the Geometric Average Formula will be

$$GM = (x_1 \times x_2 \times x_3 \times \cdots \times x_n)^{\frac{1}{n} }$$

Geometric Mean Example

Suppose we have the following set of values $x=32, 36, 36, 37, 39, 41, 45, 46, 48$. The Computation of Geometric Mean will be

\begin{align*}
GM &= (32\times 36 \times 36 \tmies 37 \times 39 \times 41 \times 45 \times 46 \times 48)^{\frac{1}{9}}\\
&=(243790484520960)^{\frac{1}{9}} = 39.7
\end{align*}

For a large number of observations one can compute the GM by taking the log of all observations using the following formula:

$$GM = antilog \left[\frac{\sum\limits_{i=1}^n log\, x}{n} \right]$$

$x$	$log\, x$
32	Log 32 = 1.5051
36	log 36 = 1.5563
36	log 36 = 1.5563
37	log 37 = 1.5682
39	log 39 = 1.5911
41	log 41 = 1.6128
45	log 45 = 1.6532
46	log 46 = 1.6628
48	log 48 = 1.6812
Total	14.3870

\begin{align*}
GM &= antilog \left[ \frac{\sum\limits_{i=1}^n log\, x}{n} \right]\\
&= antilog \left[\frac{14.3870}{9}\right] = antilog [1.5986]\\
&= 38.7
\end{align*}

One important point that should be remembered is that if any value in the data set is zero or negative then the GM cannot be computed.

Geometric Mean for Grouped Data

The GM for grouped data can also be computed using the following formula:

$$GM = antilog \left[ \frac{\Sigma f\times log\, x}{\Sigma f} \right]$$

Suppose, we have the following frequency distribution as follows:

Classes	Frequency
65 to 84	9
85 to 104	10
105 to 124	17
125 to 144	10
145 to 164	5
165 to 184	4
185 to 204	5
Tota	60

The GM of the above frequency distribution can be performed as follows

\begin{align*}
GM &= antilog \left[ \frac{124.2471}{60} \right]\\
&=antilog (2.0708) = 117.4
\end{align*}

The GM is particularly useful when dealing with rates of change or ratios, such as growth rates in investments. That is because geometric mean considers how things are multiplied over time, rather than simply added.

Use and Application of Geometric Mean

Geometric Mean is useful in situations like:

• Investment returns: When one looks at average investment growth, one wants to consider how much one’s money is multiplied over time, not just the change each year. That is why the GM is better suited for this scenario.
• Rates of change: Similar to investment returns, if something is increasing or decreasing by a percentage each time, the GM is a more accurate measure of the overall change.
• Growth Rates: When dealing with percentages or ratios that change over time (like investment returns or population growth), the geometric mean provides a more accurate picture of the overall change compared to the arithmetic mean.
• Proportional Changes: It is helpful for situations where changes are multiplied, not added. For example, if a recipe calls for doubling all ingredients, the geometric mean of the original quantities represents the final amount.

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