What is Harmonic Mean?

The harmonic mean is the reciprocal of the arithmetic mean of reciprocals, i.e., the average 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

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  • One can see it’s the reciprocal of the normal mean.The harmonic mean for the normal mean is ∑ x / n, so if the formula reverses, it becomes n / ∑x. So then all the values of the denominator that one must use 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 reciprocal values.The derived value would always be less than the average or the arithmetic meanArithmetic MeanArithmetic mean denotes the average of all the observations of a data series. It is the aggregate of all the values in a data set divided by the total count of the observations.read more.

Examples

Example #1

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

Solution:

Use the following data for the calculation.

The Harmonic mean =  n / ∑ [1/ Xi ]

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

= 4 / 0.99

Example #2

Mr.Vijay is a stock analyst at 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 a market capitalization of $2 billion. Company X reports $3 billion and a market capitalization of $9 billion; Company Y reports earnings of $10 billion and a market capitalization of $40 billion. Next, calculate the harmonic mean for the P/E ratio of the index.

First, we shall calculate the P/E ratioP/E RatioThe price to earnings (PE) ratio measures the relative value of the corporate stocks, i.e., whether it is undervalued or overvalued. It is calculated as the proportion of the current price per share to the earnings per share. read more.

P/E ratio is essentially (the market capitalizationMarket CapitalizationMarket capitalization is the market value of a company’s outstanding shares. It is computed as the product of the total number of outstanding shares and the price of each share.read more / the earnings).

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

Calculation of 1/X value

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

The calculation can be done as follows,

The Harmonic mean = n / ∑[1/ Xi]

  • =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 1st half of the journey and 70 mph for 2nd half from his home to the beach. What will be his average speed?

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

The calculation is as follows,

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

  • =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, have several usages. They are mainly used in finance for certain average data, such as price multiples. For example, financial multiples like the P/E ratio must not be averaged using the normal mean or the arithmetic mean because those mean are biased towards the larger values. One can also use harmonic means to identify a certain type of pattern like Fibonacci sequences that market technicians use in technical analysis.

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

The harmonic mean is defined rigidly and is based upon all values or observations in a given dataset or sample, and it can be suitable for further mathematical treatment. Like the geometric mean, the Harmonic mean is also not affected much by the observations or sampling fluctuations. It 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.

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

  • MEDIAN Formula ExcelMEDIAN Formula ExcelMEDIAN function in Excel gives the median of a given set of numbers. MEDIAN Identifies the location of the center of a group of numbers in a statistical distribution.read moreFormulaFormulaThe Weighted Mean equation is a statistical method determining the average by multiplying the weights with their respective mean and taking its sum. It is an average in which weights are assigned to individual values determining each observation’s relative importance. Weighted Mean = ∑ni=1 (xi*wi)/∑ni=1wi
  • read more of Weighted Mean Mean vs. MedianMean Vs. MedianMean is an average of given numbers. It sums up the numbers and divides them with the count of numbers which provides us with the mean. On the other hand, the median returns the middle number from the whole data set.read moreMedian FormulaMedian FormulaThe median formula in statistics is used to determine the middle number in a data set that is arranged in ascending order. Median ={(n+1)/2}thread more