Harmonic Mean

TL;DR

An average computed from the reciprocals of values rather than the values themselves.
Gives more influence to smaller values and thus reflects slower rates or lower scores more strongly.
Useful when averaging rates or when extremes (very high or very low values) should affect the central tendency.

Definition

Harmonic mean is a type of average used to calculate the central tendency of a set of numbers. It is calculated by taking the reciprocal of each number in the set, adding them together, and then taking the reciprocal of the sum.

Explanation

The harmonic mean aggregates values by converting each to its reciprocal, combining those reciprocals, and converting back by taking the reciprocal of the total. Because this process emphasizes smaller original values (their reciprocals are larger), the harmonic mean reflects the impact of lower values more strongly than the arithmetic mean or median.

Examples

Average speed

When calculating the average speed of a vehicle over a certain distance, compute the reciprocal of each speed measurement, add those reciprocals, and then take the reciprocal of that sum. This accounts for slower speeds as well as faster speeds and gives a more accurate representation of average speed.

Average grade

When calculating the average grade in a class, compute the reciprocal of each student’s grade, add those reciprocals, and then take the reciprocal of that sum. This method accounts for lower grades as well as higher grades and can give a more accurate representation of the class average.

Use cases

Harmonic mean is useful in situations where it is important to consider the extremes of a set of numbers, rather than just the average or the median. It can provide a more accurate representation of the central tendency in a set of numbers, and is particularly useful in situations where there are a few extremely high or low values.

Average
Median