Harmonic Mean

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

• 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:

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)

Why

In some rate type questions the harmonic mean gives the true answer!

Example: we travel 10 km at 60 km/h, then another 10 km at 20 km/h, what is our average speed?

Harmonic mean = 2/( 160 + 120 ) = 30 km/h

Check: the 10 km at 60 km/h takes 10 minutes, the 10 km at 20 km/h takes 30 minutes, so the total 20 km takes 40 minutes, which is 30 km per hour

The harmonic mean is also good at handling large outliers.

Example: 2, 4, 6 and 100

The arithmetic mean is 2+4+6+1004 = 28

The harmonic mean is 4/( 12 + 14 + 16 + 1100 ) = 4.32 (to 2 places)

But small outliers will make things worse!

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 1s on the other, and the mean on one side matched with the values on the other side too).