Muhammad Imdad Ullah

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:

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

\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:

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

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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Introduction to Weighted Averages

The multipliers or sets of numbers that express more or less relative importance of various observations (data points) in a data set are called weights.

The weighted arithmetic mean (simply called weighted average or weighted mean) is similar to an ordinary arithmetic mean except that instead of each data point contributing equally to the final average, some data points contribute more than others. Weighted means are useful in a wide variety of scenarios. Weighted averages are used when there are a bunch of values, but some of those values are more important or contribute more to the overall result.

Example of Weighted Average

For example, a student may use a weighted mean to calculate his/her percentage grade in a course. In such an example, the student would multiply the weight of all assessment items in the course (e.g., assignments, exams, sessionals, quizzes, projects, etc.) by the respective grade that was obtained in each of the categories.

As an example, suppose in a course there are a total of 60 marks, while the distribution of marks is as follows, Assignment-1 has a weightage of 10%, Assignment-2 has a weightage of 10%, the mid-term examination has a weightage of 30% and the final term examination have the weightage of 50%. The scenario is described in the table below:

Weighted Average Formula

Mathematically, the weighted average forma is given as

$$\overline{x}_w = \frac{\sum\limits_{i=1}^n w_i x_i}{\sum\limits_{i=1}^n w_i}$$

Another Example

Consider another example: Suppose we have monthly expenditures of a family on different items with their quantity

The average expenses will be: $AM = \frac{590}{5} = 118$.

However, the weighted average of the scenario will be $\overline{x}_w = \frac{\sum\limits_{i=1}^n w_i x_i}{\sum\limits_{i=1}^n w_i} = \frac{2539.5}{12.5}=203.16$

Keeping in mind the importance of weight, the average monthly expenses of a family was 203.16, not 118.

Note that in a frequency distribution, the computation of relative frequency (rf) is also related to the concept of weighted averages.

Some Real-World Examples of Weighted Averages

• Calculating class grade: Different assignments might have different weights (e.g., exams worth more than quizzes). A weighted mean considers these weights to determine the overall grade.
• Stock market performance: A stock index might use a weighted average to reflect the influence of large companies compared to smaller ones.
• Customer Satisfaction: Finding the average customer satisfaction score when some customers’ feedback might hold more weight (e.g., frequent buyers).
• Average Customer Spending: if some customers buy more frequently.
• Expected Value: Determining the expected value of outcomes with different probabilities.

The following are some important questions. What is the importance of weighted mean? Describe its advantages and disadvantages. What is an average? What are the qualities of a good average? What does Arithmetic mean? Describe the advantages and disadvantages of Arithmetic mean. In which situations do we apply arithmetic mean?

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