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:
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).