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

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

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

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

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

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:

65 to 849
85 to 10410
105 to 12417
125 to 14410
145 to 1645
165 to 1844
185 to 2045

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

Classes$f$$X$$log\, X$$f \times log\, X$
65-84974.5log 74.5 = 1.872216.8494
85-1041094.5log 94.5 = 1.975419.7543
105-12417114.5log 114.5 = 2.058834.9997
125-14410134.5log 134.5 = 2.128721.2872
145-1645154.5log 154.5 = 2.188910.9446
165-1844174.5log 174.5 = 2.24188.9672
185-2045194.5log 194.5 = 2.288911.4446
Total60  124.2471

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

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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