Harmonic Mean

Reference > Mathematics > Statistics > Pythagorean Means

In addition to the commonly used arithmetic mean (what we commonly refer to as the "average") we also have something called a harmonic mean. The harmonic mean is a little bit more cumbersome to calculate, as shown by the formula below.

H =

a₁

a₂

a₃

+ ... +

a_n

This is a bit overwhelming, since it is designed to represent the harmonic mean of n data points. So let's pare it down to just two data points.

Example
What is the harmonic mean of 4 and 6?

Solution
H =

H =

= 4.8

Like the arithmetic mean, the harmonic mean will always like somewhere between the smallest and largest numbers being averaged. However, it will be weighted toward the lower end of the values.

Example
What is the harmonic mean of 3, 6, and 9?

Solution
H =

H =

= 5.4

When to Use
The harmonic mean can be used in science to deal with outliers. Outliers are data points which are so out-of-line with the rest of the data that they seem silly. Since the harmonic mean is weighted toward the lower values, using the harmonic mean lets you use that data point without having it skew the results too badly.

For example, suppose you are my student, and your test grades are 60, 58, 62, 68, and 100. If I'm a nice teacher, I'll just use the arithmetic mean to get your average:

A =

60 + 58 + 62 + 68 + 100

= 69.6, which rounds up to a 70.

On the other hand, if I was a terribly unkind teacher, I might look at that and say, "That 100 is ridiculous! Probably he cheated, and I'm not going to count that grade!"

A =

60 + 58 + 62 + 68

= 62, which isn't nearly as nice.

Or, instead of being terribly unkind, I could just be a little "mean." Here I use the harmonic mean, which counts the 100, but it doesn't have as much weight:

H =

100

H = 66.9, which rounds up to 67. It's better than dropping the outlier altogether, but it's not as nice to the student as using the arithmetic mean.

Of course, I don't know any teachers who actually do this to their students, but in science it's a nice way of keeping outliers in your calculations without letting them affect your results too badly.

A word of caution, though: it doesn't work the other way - if the outlier is low, don't use the harmonic mean, or your average will be skewed toward the outlier! Only use the harmonic mean if you have outliers that are on the large side.