The harmonic mean is:
the reciprocal of the average of the reciprocals
Yes, that is a lot of reciprocals!
Reciprocal just means 1value.
The formula is:
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)
In some rate type questions the harmonic mean gives the true answer!
Example: we travel 10 km at 60 km/h, than 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).