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Harmonic Mean

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.

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Harmonic Mean -

Tag(s) : Statistics

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# Harmonic Mean

## Harmonic Mean Calculator

### How to compute an harmonic mean? (Definition)

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:

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

### How to compute an harmonic mean?

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.

### When to use an harmonic mean?

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.

### What is the harmonic series? (Definition)

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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