# Harmonic Mean

The harmonic mean is:

the reciprocal of the arithmetic mean of the reciprocals

Yes, that is a lot of reciprocals!

("Reciprocal" just means \frac{1}{value})

The formula is:

Where **a,b,c,...** are the values, and **n** is how many values.

Steps:

- Calculate the reciprocal (1/value) for every value.
- Find the average of those reciprocals (just add them and divide by how many there are)
- Then do the reciprocal of that average (=1/average)

### Example: What is the harmonic mean of 1, 2 and 4?

The reciprocals of 1, 2 and 4 are:

\frac{1}{1} = 1, \frac{1}{2} = 0.5, \frac{1}{4} = 0.25

Now add them up:

1 + 0.5 + 0.25 = 1.75

Divide by how many:

Average = \frac{1.75}{3}

The reciprocal of that average is our answer:

Harmonic Mean = \frac{3}{1.75} = **1.714** (to 3 places)

## Another way to think of it

We can rearrange the formula above to look like this:

It is *not* easy to use this way, but it does look more "balanced" (**n** on one side matched with n **1**s on the other, and the mean matched with the values too).