Tool for calculating a harmonic mean from a series or list of integers or real numbers. The harmonic mean is for example used for average speeds.

Harmonic Mean - dCode

Tag(s) : Statistics

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

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

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

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