Harmonic Mean
The harmonic mean is calculated as the reciprocal of the arithmetic mean of the reciprocals of a given set of values: $$ \mu_h = \frac{n}{\sum_{i=1}^n \frac{1}{x_i} } = \frac{n}{ \frac{1}{x_1} + \frac{1}{x_2} + ... + \frac{1}{x_n} } $$
What is it used for?
The harmonic mean is a type of central tendency measure.
It does not yield the same value as the arithmetic mean.
When should you use the harmonic mean? The harmonic mean is ideal for situations involving quantities that are inversely proportional, or when dealing with ratios (e.g., speed). It's also appropriate when the data is arranged in an arithmetic progression, meaning there is a constant difference between consecutive terms. Additionally, it is frequently used to calculate the purchasing power of money, as it considers the reciprocal of prices.
A Practical Example
Consider the following set of data:
$$ 1 \ , \ 5 \ , \ 7 \ , \ 3 \ , \ 6 \ , \ 8 $$
Here, n = 6 because there are 6 values in the dataset.
The harmonic mean of this set is approximately 3.04.
$$ \mu_h = \frac{ 6 }{ \frac{1}{1} + \frac{1}{5} + \frac{1}{7} + \frac{1}{3} + \frac{1}{6} + \frac{1}{8} } $$
$$ \mu_h = \frac{ 6 }{ 1.97} $$
$$ \mu_h = 3.04 $$
Example 2
Let's look at the speeds of two cars:
$$ x_1 = 60 \ km/h $$
$$ x_2 = 120 \ km/h $$
These speeds are defined as ratios (distance/time).
The harmonic mean of these two speeds is 80 km/h.
$$ \mu_h = \frac{2}{ \frac{1}{60} + \frac{1}{120} } = \frac{2}{ \frac{3}{120} } = 2 \cdot \frac{120}{3} = 80 \ km/h $$
Note: The arithmetic mean of these two speeds is 90 km/h: $$ \mu = \frac{60+120}{2} = \frac{180}{2} = 90 \ km/h $$
Key Points
Here are some important considerations regarding the harmonic mean:
- All values must be non-zero. When calculating the harmonic mean, all values in the set must be non-zero. Otherwise, the reciprocal of a zero value would result in division by zero, which is undefined.
- Weighted Harmonic Mean
The weighted harmonic mean is a variation where each value is given a specific weight (wi): $$ \mu = \frac{ \sum w_i}{ \sum \frac{1}{x_i} \cdot w_i} $$
And so on.
