The harmonic mean of two numbers is equal to the reciprocal of the arithmetic mean of the reciprocals of those numbers.
In simpler terms, if you have two numbers, let's call them 'a' and 'b', the harmonic mean (often denoted as 'H') is calculated as follows:
- Find the reciprocals: Calculate 1/a and 1/b.
- Calculate the arithmetic mean of the reciprocals: Add the reciprocals together (1/a + 1/b) and divide by 2: (1/a + 1/b) / 2
- Find the reciprocal of the arithmetic mean: Take the reciprocal of the result from the previous step: 1 / [(1/a + 1/b) / 2]
This can be simplified to the following formula:
H = 2 / ( (1/a) + (1/b) )
Or, further simplified to:
H = 2ab / (a + b)
Example:
Let's find the harmonic mean of the numbers 4 and 6.
-
Using the formula H = 2 / ( (1/a) + (1/b) ):
H = 2 / ( (1/4) + (1/6) )
H = 2 / ( (3/12) + (2/12) )
H = 2 / (5/12)
H = 2 * (12/5)
H = 24/5
H = 4.8 -
Using the formula H = 2ab / (a + b):
H = (2 4 6) / (4 + 6)
H = 48 / 10
H = 4.8
Therefore, the harmonic mean of 4 and 6 is 4.8.
The harmonic mean is often used in situations involving rates or ratios. It gives a different type of average than the arithmetic mean and is particularly useful when dealing with problems where the rates are inverted or represent speeds over the same distance.
