The mean is:

the reciprocal of the first moment of the reciprocals

Yes, that’s tons of reciprocals!

here a,b,c,… are the values, and n is what percentage values.


Calculate the reciprocal (1/value) for each value.

Find the typical of these reciprocals (just add them and divide by what percentage there are)

Then do the reciprocal of that average (=1/average)

Example: what’s the mean of 1, 2 and 4?

The reciprocals of 1, 2 and 4 are:

11 = 1, 12 = 0.5, 14 = 0.25

Now add them up:

1 + 0.5 + 0.25 = 1.75

Divide by how many:

Average = 1.753

The reciprocal of that average is our answer:

Harmonic Mean = 31.75 = 1.714 (to 3 places)

Another way to consider it

We can rearrange the formula above to seem like this:

Harmonic Mean

It is tough to use this manner , but it does look more “balanced” (n on one side matched with n 1s on the opposite , and therefore the mean matched with the values too).