Geometric Mean Formula

Introduction to the 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 (a useful measure of central tendency) 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, 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 the Geometric Mean will be

\begin{align*}
GM &= (32\times 36 \times 36 \times 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 the geometric mean considers how things are multiplied over time rather than simply added.

Use and Application of the Geometric Mean

The GM 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: This 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.

Real-Life Examples

1. Finance (Average Investment Returns): To calculate the average rate of return on investments over time one can use the Geometric Mean? It is because the returns are compound that one cannot use the arithmetic mean. For example, Year 1 return = +10%, the Year 2 return = -20%, Year 3 return = +30%, the GM Return will give the true average annual return over 3 years.
2. Economics (Growth Rates): The GM should be used to compute average GDP growth, inflation, or population growth over multiple years. It is because growth over time is multiplicative. For example, GDP grows at 3%, 4%, and 5% over 3 years. The geometric mean provides the average annual growth rate.
3. Business (Average Rate of Change in Prices or Sales): To find the average percentage change in prices or sales across several periods, the GM can be used. For example, A product price increased by 10%, then decreased by 5%, then increased by 8%. The GM will give the true average percentage change.
4. Environmental Science (Air or Water Quality Data): The GM should be used to calculate the average concentration of pollutants, as environmental data often contains highly skewed values. For example, Pollution levels: 2, 4, 8, 50 → The arithmetic mean is skewed by 50, therefore, the Geometric Mean will give a better central tendency for such data.
5. Demographics (Fertility or Mortality Rates): In demographic research, to average birth or death rates across different countries or regions, The Geometric Mean should be used because these rates are often ratios and vary widely between groups.
6. Health & Medicine (Drug Dosage and Bacterial Growth): To measure average bacterial growth rates, enzyme activities, or dosage effectiveness. It is because these processes grow or decline exponentially.
7. Marketing (Average Performance Metrics): The Geometric Mean should be used to calculate the average conversion rate or engagement rate over multiple platforms or campaigns. Because metrics are often multiplicative percentages, and geometric mean gives a more accurate reflection.

Summary Table

Field	Application	Why Use Geometric Mean?
Finance	Investment returns	Captures compound interest effects
Economics	GDP, inflation, population growth	Works well with growth rates
Business	Sales/prices over time	Accurately represents percent changes
Environment	Pollution concentration levels	Handles skewed environmental data
Demographics	Birth/death rates	Compares ratios fairly
Medicine	Bacterial/drug growth rates	Models exponential biological change
Marketing	Engagement/conversion rates	Why Use the Geometric Mean?

Summary

The geometric mean is incredibly useful in real-life situations where values are multiplied together, grow exponentially, or vary in ratios or percentages — rather than being added. The GM is very useful, especially in business, finance, science, and data analysis.

FAQS about Geometric Mean

• What is meant by the Geometric Mean?
• In what situation should the GM be used?
• For what observations, the GM not computed?
• Write down the formula of GM for group and ungrouped data.
• Give some real-life examples that make use of the Geometric Mean.

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