Harmonic Mean

With a list of n values \( X = \{x_1, x_2, \dots, x_n \} \). The harmonic mean is defined by the ratio of n to the sum of the inverse of the values. $$ \bar{x}_{harm} = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}} $$

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 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 resistances in parallel corresponds to the harmonic average of the 2 resistance values.