What is Harmonic Mean in statistics & It’s formula?

Harmonic mean is defined as the reciprocal of the average of reciprocal of the values.

In statistics, data science or just general everyday decisions, we always are interested in knowing the middle value.

Like what’re the average marks, average speed or average returns etc.

In technical terms, we also call it central tendency.

I will try to make this article as practical and as easy as possible. So hang on and read till the end.

Table of Contents

  1. Introduction to Harmonic Mean
  2. Harmonic Mean Formula
  3. Relationship: AM,HM,GM
  4. Uses of Harmonic Mean
  5. Examples of Harmonic Mean
  6. Calculating HM in excel
  7. Advantages and Disadvantages

Introduction to Harmonic Mean

The harmonic mean is a method which gives less weightage to larger single values and more weightage to smaller values.

I want you to directly look at this example to understand this. We have two sets of students marks, Class A and Class B.

Class A has one extremely large value of 100, so harmonic mean actually gave lower weightage.

In Contrast, Class B values have relatively smaller values, hence more or less the harmonic mean is closer to the arithmetic mean.

So what’s important for you to remember is, when is it used?

When the data has larger extreme values.

Harmonic Mean Formula

I will explain this in two parts, one is give you the logic of investing the value and second on why we don’t take the average directly.

The Formula

,wherere x1 x2 x3 xn are individual values and n is the total number of observations.

Explaination of the Formula

Don’t memorise the formula, let me make you understand the logic instead.

Why Reciprocal?

Take for an example we have two values; 50 and 100.

Let me put up two questions to you;

  • Which is larger? 50 or 100? ofcourse 100 but,
  • 1/50 and 1/100, now which is larger? the answer now changes to 1/50.

Hence the reason to take the reciprocal of values, even if the values are large. The reciprocal will be small.

Another question you might ask is:

Why do we divide N by the reciprocals?

Well, again lets see with our previous example.

If I just sum all the reciprocal and divide by number of observations, in our case 4. Then I get a weird answer.

Instead, if I divide N, in our case 4 with the sum of reciprocals then I get the right answer.


The Relationship of Harmonic Mean ,Arithmetic Mean, Geometric Mean

The above three means are also called as pythogoream means.

You might be wondering, am I going to make this complicated? Absolutely not.

However, lets try to play a little with numbers.

  • Arithemetic Mean: a +b /2
  • Geometric Mean:√ab
  • Harmonic Mean: 2/(1/a+ 1/b) or 2 ab/(a+b)

So we can replace 2ab with geometric mean square, because GM is √ab.

  • Therfore, harmonic mean = 2 (GM)²/(a+b)

but 2/(a+b) can also be said to reverse of AM.

Hence, Harmonic Mean= (GM)²/AM

Uses of Harmonic Mean

Given below is some of the popular uses of harmonic mean

  • It can be used in speed calculation, because speed is already a ratio of distance over time.
  • Heavily used in finance when dealing with multiples
  • It can be used to finance patterns in fibonacci series
  • Very helpful when dealing with data, which has larger extreme values

Weighted Harmonic Mean

This is just a special case, where along with finding the mean, we also wish to assign weightage.

However all the weights, should add up to 1.

weighted harmonic mean

So, all that changed from our previous formula, instead of just reciprocal. We know are dividing it by the weightage.

An unweighted harmonic mean(HM), assigns equal weightage to all values right?

However, if you wish to assign weightage then you could use this method.

Just to add here, this is not a special case of just harmonic mean but you could also find weighted arithmetic mean too.

Why Do We Calculate Mean?

Consider that you are evaluating a mutual fund investment scheme. Which one of the below questions will you be interested in asking?

1.What was the highest return for a particular month?

2.Or what has been the average returns?

Lets take the example on this sheet: Franklin Templeton MF.

Average

Median

0.58%

1.02%

After the analysis we concluded that the average monthly return is:

  • Does the central location itself give you any information?
  • What if we were to find how much does each day fluctuate from the centre?
  • Because we are always interested in the average fluctuation, which will somehow affect our capital volatility.
  • Imagine investing in a fund and even though the returns are good at 21% annual
  • Just when you were planning to buy a house the fund experiences a -70% fluctuation
  • This is why we are learning statistics and this is why managing risk is more important than targeting returns.

Harmonic Mean Examples

Example -1

You have to find the harmonic mean of the average P/E ratio of the index of the stocks of Company A and Company B.

  • Company A has a market capitalization of US$1 billion and earnings of US$20 million
  • Company B has a market capitalization of US$20 billion and earnings of US$5 billion.
  • The index comprises 40% of Company A and 60% of Company B. 

We first need to calculate the P/E of each company.

The calculation can be done in the following way:

  • P/E (Company A) = ($1 billion) / ($20 million) = 50
  • P/E (Company B) = ($20 billion) / ($5 billion) = 4

We must use the weighted harmonic mean to calculate the P/E ratio of the Index.

Using the formula for the weighted harmonic mean, the P/E ratio of the index can be found:

P/E(Index) = (0.4+0.6) / (0.4/50 + 0.6/4) = 6.33

Example-2

This central tendency method is particularly useful for rates or ratio. For example look at the harmonic mean of P/E Ratios of Nifty

What this does is, the exact opposite of weighted mean.

The higher the value the lower the proportion.

If you had calculate the average rate of P/E, how would you calculate it? You would have got 30 and if you had calculate the Weighted mean you would have got 32.

P/E Ratios- nifty

Inverse Value(1/x)

25

4%

35

3%

40

3%

20

5%

Sum of Inverse

14%

HM (no of obs/ sum of inverse)

27.86

How to calculate Harmonic Mean in Excel?

Calculating harmonic mean in excel is very easy, because it saves the trouble of mannually finding the inverse using the harmonic mean.

God bless bill gates for creating spreadsheets!

However a word of caution, the harmonic mean doesn’t work with negative or zero values.

You can practice this in this template:


Advantages of Harmonic Mean

While we seen how to calculate this, and at the same time learnt when to use it.

Now its time to look its advantages:

  • Its approporate for time and rate calculation
  • It can be usefull for derived mathematical calculation
  • Harmonic mean isn’t effect by the outliers

Disadvantages of Harmonic Mean

Ofcourse nothing comes without some shortcomings, and this method likewise has some challenges.

  • The method is definetely complex
  • It gets affected by extreme values, in opposite situations of arthemetic mean.

Conclusion

Central tendency is a large and wide concept, having its application in modern day machine learning as well.

It is important to understand, the core objective of calculating central tendency, without which you can make some serious errors with your decisions. Imaging that you are taking a decision to invest $100 Mn in an investment opportunity, while concluding its average risk.

Selecting an in appropriate method of calculation, could mean disaster. Similarly, imagine you are trying to measure the chance of failure for a business project, and we causally look at a skewed sample.

The skewed sample might contain data of only successful projects and not the failures.

Moreover it is also possible that the data has a time bias, which was particularly favorable for this project. On the contrary this could also be true with a data set that contains only failures.

You see, its not about calculating a value, its about what exactly do we intent do with that value?

We are in an education system that casually deals with concepts and fundas, without any heed to its real world relevance. Hence its important that as students of maths, finance or science you push beyond what the curriculum draws on.

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ALLEN ARAVINDAN,CFA

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