The harmonic mean is:
the reciprocal of the arithmetic mean 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)
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 matched with the values too).