Harmonic mean

The harmonic mean (H) of n numbers ( x1, x2, x3, ... , xn ), also called subcontrary mean, is given by the formula below.
Harmonic mean

If n is the number of numbers, it is found by dividing the number of numbers by the reciprocal of each number.


Suppose there are two numbers.
H =
2 / 1/x1 + 1/x2

Suppose there are 3 numbers.
H =
3 / 1/x1 + 1/x2 + 1/x3

Examples showing how to calculate the harmonic mean

Example #1:

Find the harmonic mean of 3 and 4

H =
2 / 1/3 + 1/4


H =
2 / 4/12 + 3/12


H =
2 / 7/12


H =
2 × 12 / 7


H =
24 / 7
= 3.4285


Example #2:

Find the harmonic mean of 1, 2, 4, and 10

H =
4 / 1/1 + 1/2 + 1/4 + 1/10


H =
4 / 20/20 + 10/20 + 5/20 + 2/20


H =
4 / 37/20


H =
4 × 20 / 37


H =
80 / 37
= 2.1621

A linear motion problem that leads to the harmonic formula.

A car travels with a speed of 40 miles per hour for the first half of the way. Then, the car travels with a speed of 60 miles per hour for the second half of the way. What is the average speed? 

Average speed =
total distance / total time

First notice that it is not possible to use directly the speed formula since we do not know for how long the car kept driving with a speed of 40 m/h and then 60 m/h. However, with some manipulation, we can still tackle the problem.

Let t1 be the time it took to travel the first half of the total distance

Let d be the first half of the total distance.

t1 =
d / 40


Let t2 be the time it took to travel the second half of the total distance

Let d be the second half of the total distance.

t2 =
d / 60


Total time = t1 + t2 = d/40 + d/60

Total distance = d + d = 2d

Now replace these in the formula

Average speed =
total distance / total time


Average speed =
2d / d/40 + d/60


Average speed =
2d / d(1/40 + 1/60)


Cancel d and the average speed =
2 / (1/40 + 1/60)


Now, you can see that it looks like we are calculating the harmonic mean for 2 numbers by using the formula above.

H = average speed =
2 / (3/120 + 2/120)


H = average speed =
2 / (5/120)


H = average speed =
2 × 120 / 5


H = average speed =
240 / 5
= 48 miles per hour

See an example of harmonic mean related to the stock market 

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