Quantitative Methods Used in Salary Administration

Geometric mean

The arithmetic mean is useful when considering average salaries for positions since the observation values are independent.  However, it is not as useful when you need to look at trends in the average salary for a particular position over time and values from one year to the next are dependent.

Example: You know that the salary increases for year 1, 2 and 3 are 10%, 15% and 20% respectively. To find out the average salary increase over 3 years, it is incorrect to simply add the percentage increases together and divide by 3.

10% + 15% + 20%
3

This gives you the arithmetic mean. It does not give you a true measure of the average percent increase over the 3 years because the salary increase for the first year was 10% of the previous year, not the previous year's salary plus 10. Therefore, when there is some dependency among observation values, you must use the geometric mean.

The geometric mean can be computed using the following formula:

Geometric mean = ( ⁿ√(X₁X₂...X_n) ) - 1

The percentages are converted to their decimal values and added to 1.

In this case, the geometric mean would be:

Geometric mean = ( ³√(1.10 x 1.15 x 1.20) ) - 1 = 0.149

Multiplying by 100 we get a geometric mean of 14.9%

Note that had you used the arithmetic mean, your answer would have been overly optimistic. The average salary increase over the 3 years is 14.9%, not 15%.

Computing geometric mean for negative percentages

Imagine if starting salaries in your area increased 10% over one year, increased 20% over the following year, and decreased 15% during the third year. To find the average increase over the three years, you must first convert each percentage as follows:

The Year 1 increase is 10% or 0.10. Adding 1 this converts to 1.10.

The Year 2 increase is 20% or 0.20. Adding 1 this converts to 1.20.

The Year 3 increase is -15% or -0.15. Adding 1 this converts to 0.85.

Next, apply the geometric formula.

( ³√(1.10 x 1.20 x 0.85) ) - 1 = 0.0391

Then, convert the result back to a percentage by multiplying 0.0391 x 100. Thus, the average increase is 3.91%.

The geometric mean will always be lower than the arithmetic mean for the same set of observations.

(Additional information on geometric mean.)

Example of geometric mean

A survey reports that overall salary increases were:

What is the average salary increase from 2020 to 2022?

Since this problem deals with percentage increases over time, we need to use the geometric mean to calculate the average salary increase from 2020 to 2022.

Geometric mean = ( ⁿ√(X₁X₂...X_n) ) - 1

Geometric mean = ( ³√(1.04 x 1.08 x 1.12) ) - 1 = 0.0795

Multiply 0.0795 by 100 to get 7.95%

The average salary increase from 2020 to 2022 is 7.95%.

As noted above, you would get a higher average if you use the arithmetic mean method:

(4% + 8% + 12%) / 3 = 8.0%