Harmonic Mean

With a list $ X $ of n values/numbers $ \{x_1, x_2, \dots, x_n \} $. The harmonic mean is defined by the ratio of $ n $ to the sum of the inverse of the values/numbers.

$$ \bar{x}_{harm} = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}} $$

To compute a harmonic mean of a list of values, count the total number $ n $ of values in the list and calculate the sum $ S $ of the inverse values.

Example: A car drove a distance $ d $ at 30km/h half the distance then to 90km/h. The average speed of the car can be defined with its harmonic mean speed by the calculation $ n/S $ with $ n = 2 $ and $ S = 1/30 + 1/90 = 0.0444... $ so $ \bar{M}_{harm} = 2/(1/30+1/90) = 45 $ km/h.
Indeed, taking the distance $ d = 15km $, the car will have traveled $ d/2 $ at 30km/h in 15 minutes and $ d/2 $ at 90km/h in 5 minutes, so a total distance of 15km in 20 minutes or 45km/h on average.

The harmonic mean is used when the compared elements have inverse proportionality ratios.

Example: The price per square meter of a house is higher if the total area is small.

Example: Travel time is shorter when the speed is high.

Example: On an electronic circuit, the calculation of two resistors in parallel corresponds to the harmonic mean of the 2 resistors' values.

The harmonic series is the sequence of inverses of non-zero natural numbers denoted $ H_n $

$$ H_n = 1 + \frac12 + \frac13 + \frac14 + \cdots + \frac1n = \sum_{k=1}^n \frac1k $$