# Harmonic mean

Harmonic means are a type of mean. It is the number of values divided by the reciprocal of the values.[1] If there are ${\displaystyle N}$ numbers ${\displaystyle X_{1},X_{2},X_{3}...X_{N}}$, then the harmonic mean of these numbers are

${\displaystyle {\frac {N}{{\frac {1}{X_{1}}}+{\frac {1}{X_{2}}}+{\frac {1}{X_{3}}}+\dots +{\frac {1}{X_{N}}}}}}$

Out of the geometric mean and arithmetic mean, the harmonic mean is usually the smallest.[2]

## Example

Let's find the harmonic mean of 2,4 and 5. There are three numbers so we will be dividing three. The reciprocals of the numbers are ${\displaystyle {\frac {1}{2}}}$, ${\displaystyle {\frac {1}{4}}}$ and ${\displaystyle {\frac {1}{5}}}$. If we add the reciprocals we get ${\displaystyle {\frac {19}{20}}}$. If we divide three by this number, the result is ${\displaystyle {\frac {60}{19}}}$ (approximately 3.157894737)

## References

1. "mean | Definition, Formula, & Facts | Britannica". www.britannica.com. Retrieved 2022-08-26.
2. Weisstein, Eric W. "Pythagorean Means". mathworld.wolfram.com. Retrieved 2022-08-26.