## What Is Harmonic Mean Formula?

*A harmonic mean formula is an important mathematical concept that is widely used to calculate what is the average of a data set. In the financial market, it is often used in cases involving ratios, rates, etc, which leads to the identification of patterns.*

You are free to use this image on your website, templates, etc, Please provide us with an attribution linkHow to Provide Attribution?Article Link to be Hyperlinked

For eg:

Source: Harmonic Mean Formula (wallstreetmojo.com)

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.

##### Table of contents

### Harmonic Mean Formula Explained

The harmonic mean formula is a method that is used to find the reciprocals of all the arithmetic means of a particular set of numerical or financial data. Situations which deals with rates, prices or ratios, are users of such a method for finding the average.

In the financial market the main users of this calculation of **harmonic mean formula for grouped data** are the analysts and investors. They can identify patterns in prices through this method and take investment decisions.

The formula for the same is given below.

**Harmonic Mean = n / ∑ [1/X**

_{i}

**]**

- 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.

###### Financial Modeling & Valuation Courses Bundle (25+ Hours Video Series)

**–>>** If you want to learn Financial Modeling & Valuation professionally , then do check this Financial Modeling & Valuation Course Bundle (**25+ hours of video tutorials with step by step McDonald’s Financial Model**). Unlock the art of financial modeling and valuation with a comprehensive course covering McDonald’s forecast methodologies, advanced valuation techniques, and financial statements.

The **harmonic mean formula for grouped data** is considered to be very useful due to the fact that it takes into account all the data or entries in a particular series and cannot be calculated if any item is eliminated. It also takes into account very small and insignificant values, which may also be negative.

However, since reciprocals is taken for each value, the entire calculation process may turn out to be complex and time taking. Moreover, a series containing a zero is very difficult to calculate because there is no reciprocal for it. So we see that this method is applicable in many cases but with some restrictions.

### Examples

Let us understand the concept of **harmonic mean formula in statistics** with the help of some suitable examples, as given below.

#### 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/ X_{i }]

= 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.**

**Solution:**

Use the following data for the calculation.

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) = 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 1 ^{st} half of the journey and 70 mph for 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 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.

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.

From the above examples we can understand how the relevant data can be extracted from financial statements of a company to calculate ratios of any other important information. The method of **harmonic mean formula in statistics** can also be used to calculate the average speed of any vehicle, etc because it takes into consideration the time that is taken in each part of the journey. In other words, the impact of extreme values can be minimised trough this method.

### 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 formula for individual series**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. - Apart from finance and economics, the method of
**harmonic mean formula for individual series**can also be used in the field of medicine or environmental science for monitoring or modelling, like the speed of flow of a river, or the average magnitude of an earthquake to provide a measure of level of seismic activity or getting health related statistics, like prevalence of a particular disease in the field of medical research. - In the field of quality control for the purpose of manufacturing, this metric is often used to calculate what is the average time taken to produce a particular product, which will help in evaluating the efficiency level of the process. It is also successfully used in data analysis and machine learning or social science.

Thus, we see from the above points that this metric has a number of uses in the modern world, which helps in getting information that are successfully implemented in various aspects of our social and economic fields.

### Recommended Articles

This article has been a guide to what is Harmonic Mean Formula. We explain the formula along with examples and its relevance and uses in various fields. 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 more
- FormulaFormulaThe 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 more
- Median 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