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Harmonic Mean-What is it? Explained with Examples


The harmonic mean is a kind of average that is based on the reciprocals of two or more numbers. It is calculated by taking the reciprocal of the arithmetic mean of the reciprocals of the given values. It is denoted by the symbol H.

Harmonic Mean Formula:

The harmonic mean for two numbers ‘a’ and ‘b’ can be calculated using the formula,

H = \frac{2}{\frac{1}{a}+\frac{1}{b}}

The formula can be simplified and stated in the form,

H = \frac{2ab}{a+b}

If we have n number of observations say x_1,x_2,\ldots,x_n, then the harmonic mean is given by the formula,

H = \frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n}}

Examples of calculating Harmonic Mean:

Example 1: Calculate the harmonic mean for the two numbers 7 and 3.

Solution: We use the formula given above,

H = \frac{2ab}{a+b} = \frac{2\times 7\times 3}{7+3} = \frac{42}{10} = \textbf{4.2}

Example 2: Calculate the harmonic mean for the three numbers 2, 3, 5, and 7.

Solution: We calculate it as follows,

H = \frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n}} = \frac{4}{\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}} = \frac{4}{1.176} = \textbf{3.4}

When to use Harmonic Mean?

The harmonic mean is restricted in its field of usefulness. it is useful for computing the average rate of increase in profits of a company or the average speed at which a journey is performed. Suppose that a man travels from point A to point B with a speed of 20 km/h and returns back from point B to point A at a speed of 30km/h. Then the average speed of the man is not given by the arithmetic mean which is 25km/h. The correct average speed is given by the harmonic mean,

\text{Average Speed } = \frac{2 \times 20 \times 30}{20+30} = \frac{1200}{50} = 24\text{km/h}

The harmonic mean cannot be computed when there are both positive and negative numbers in a series or if one or more of the numbers is zero.

Weighted Harmonic Mean:

At times it may be necessary to calculate the mean with some weights attached. For example, when calculating average speed in the above example the distance from point A to B and the return journey from B to A was equal. If the distances travelled were unequal we would have calculated the weighted harmonic mean.

It can be calculated using the formula,

H = \frac{\sum_{i=1}^{n} w_i}{\frac{w_1}{x_1}+\frac{w_2}{x_2}+\ldots+\frac{w_n}{x_n}}

where, the w_i are called the “weights”.

Harmonic Mean in Excel:

The harmonic mean can be computed in excel by using the “HARMEAN” function. Simply types the numbers in a column and select those numbers. By using the “HARMEAN” function you can obtain the desired value.

Harmonic Mean vs Geometric Mean:

In any distribution where the original numbers are different the arithmetic, geometric and harmonic means are also distinct. The arithmetic mean is greater than the geometric mean which is in turn greater than the harmonic mean. We have the following relationship,

\text{ Arithmetic Mean }\geq \text{ Geometric Mean }\geq \text{ Harmonic Mean }

This is also known as the A.M-G.M-H.M inequality.

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