## What is Harmonic Mean?

The harmonic mean is the reciprocal of the arithmetic mean of reciprocal i.e. the average is calculated by dividing the number of observations in the given dataset by the sum of its reciprocals (1/Xi) of every observation in the given dataset.

### Harmonic Mean Formula

**Harmonic Mean = n / ∑ [1/X**

_{i}

**]**

- One can see it’s the reciprocal of the normal mean.
- The Harmonic mean for normal mean is ∑ x / n so if the formula is reversed it becomes n / ∑x and then all the values of the denominator that must be used should be reciprocal i.e. for the numerator it remains “n” but for the denominator the values or the observations for them we need to use to reciprocal values.
- The value that is derived would always be less than average or say the arithmetic mean.

### Examples

#### Example #1

**Consider a data set of following numbers: 10, 2, 4, 7. Using the above-discussed formula you are required to calculate Harmonic mean.**

**Solution:**

Use the following data for the calculation.

The Harmonic mean = n / ∑ [1/ X_{i }]

= 4/ (1/10 + 1/2 + 1/4 + 1/7)

= 4 / 0.99

#### Example #2

**Mr.Vijay is a stock analyst in JP Morgan. His manager has asked him to determine the P/E ratio of the index which tracks the stock prices of Company W, Company X, and Company Y. Company W reports earnings of $40 million and the market capitalization of $2 billion, Company X reports earnings of $3 billion and the market capitalization of $9 billion and while Company Y reports earnings of $10 billion and the market capitalization of $40 billion. Calculate the Harmonic mean for the P/E ratio of the index.**

**Solution:**

Use the following data for the calculation

First, we shall calculate the P/E ratio

P/E ratio is essentially (the market capitalization / the earnings).

- P/E of (Company W) = ($2 billion) / ($40 million) = 50
- P/E of (Company X) = ($9 billion) / ($3 billion) = 3
- P/E of (Company Y) = ($40 billion) / ($10 billion) = 4

Calculation of 1/X value

- Company W = 1/50 = 0.02
- Company X= 1/3 = 0.33
- Company Y= 1/4 = 0.25

The calculation can be done as follows,

The Harmonic mean = n /__ ∑[1/ X___{i}]

- =3/(1/50 + 1/3 + 1/4)
- =3/0.60

#### Example #3

**Rey a resident of northern California is a professional sports biker and is on his tour to a beach from his home on Sunday evening around 5:00 PM EST. He drives his sports bike at 50 mph for the 1 ^{st} half of the journey and 70 mph for the 2^{nd} half from his home to the beach. What will be his average speed?**

**Solution:**

Use the following data for the calculation.

In this example, Rey went on a journey with a certain speed and here the average would be based on distance.

Calculation can be done as follows,

Here, we can calculate Harmonic mean for the average speed of Rey’s sports bike.

The Harmonic mean = n / ∑[1/ X_{i}]

- =2/ (1/50 + 1/70)
- =2/ 0.03

The average speed of Rey’s sports bike is 58.33.

### Use and Relevance

Harmonic means like other average formulas they also do have several usages and they are mainly used in the field of finance to average certain data such as price multiples. The financial multiples like P/E ratio must not be averaged using the normal mean or the arithmetic mean because those mean are biased towards the larger values Harmonic means further can also be used to identify a certain type of patterns like Fibonacci sequences that are majorly used in technical analysis by the market technicians.

The Harmonic mean also deals with averages of units such as rates, ratios or speed, etc. Also, it is important to note that it is affected by the extreme values in the given data set or in a given set of observations.

The harmonic mean is defined rigidly and is based upon all the values or all the observations in a given dataset or sample and it can be suitable for further mathematical treatment. Like the geometric mean, Harmonic mean is also not affected much with the fluctuations in observations or sampling. This would be giving greater importance to the small values or the small observations and this will be useful only when those small values or those small observations need to be given greater weight.

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This has been a guide to Harmonic Mean and its definition. Here we discuss the formula to calculate harmonic mean along with examples and downloadable excel template. You can learn more about excel modeling from the following articles –

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