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How to Calculate the Harmonic Mean

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Reviewed by Anne Schmidt

Last Updated: January 30, 2023 Fact Checked

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The harmonic mean is a way to calculate the mean, or average, of a set of numbers. Using the harmonic mean is most appropriate when the set of numbers contains outliers that might skew the result. Most people are familiar with calculating the arithmetic mean, in which the sum of values is divided by the number of values. Calculating the harmonic mean is a little more complicated. If working with a small set of numbers you may be able to solve by hand using the formula. Otherwise, you can easily use Microsoft Excel to find the harmonic mean.

Method 1

Method 1 of 3:

Setting Up the Formula

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The formula is ${\frac {n}{{\frac {1}{a_{{1}}}}+{\frac {1}{a_{{2}}}}+{\frac {1}{a_{{3}}}}+...+{\frac {1}{a_{{n}}}}}}$ , where $n$ is the number of values in the set of numbers, and $a_{{1}}$ , $a_{{2}}$ , $a_{{3}}...$ are the values in the set. ^{[1]
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Research source}
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2
Determine the values you need to find the harmonic mean for. This can be any set of numbers. ^{[2]
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Research source}
- For example, you may need to find the harmonic mean for the numbers 10, 12, 16, and 8.
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3
Plug the value of $n$ into the formula. This will equal the number of values in your set.
- For example, if you are finding the harmonic mean of the numbers 10, 12, 16, and 8, you are working with 4 values, the numerator of your formula will be 4:
  ${\frac {4}{{\frac {1}{a_{{1}}}}+{\frac {1}{a_{{2}}}}+{\frac {1}{a_{{3}}}}+...+{\frac {1}{a_{{n}}}}}}$
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4
Plug the values your are averaging into your formula. You will take the reciprocal of each number and add them in the denominator of the formula. Remember, when you take the reciprocal of a whole number, you turn the number into a fraction by placing a 1 in the numerator and the whole number in the denominator. ^{[3]
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Research source}
- For example, if the values in your set are 10, 12, 16, and 8, you would place the fractions ${\frac {1}{10}}$ , ${\frac {1}{12}}$ , ${\frac {1}{16}}$ , ${\frac {1}{8}}$ in your denominator:
  ${\frac {4}{{\frac {1}{10}}+{\frac {1}{12}}+{\frac {1}{16}}+{\frac {1}{8}}}}$

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Method 2

Method 2 of 3:

Calculating the Harmonic Mean by Hand or Calculator

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1
Add the fractions in the denominator. You can use a calculator, or add them up by hand. If you are not using a calculator, remember to find a common denominator first. To learn more about adding fractions, read Add Fractions .
- For example:
  ${\frac {4}{{\frac {1}{10}}+{\frac {1}{12}}+{\frac {1}{16}}+{\frac {1}{8}}}}={\frac {4}{{\frac {89}{240}}}}$
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2
Divide the numerator by the denominator. Remember that dividing by a fraction is the same as multiplying by its reciprocal. ^{[4]
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Research source}
- For example:
  ${\frac {4}{{\frac {89}{240}}}}$
  $=4\times {\frac {240}{89}}$
  $={\frac {4}{1}}\times {\frac {240}{89}}$
  $={\frac {960}{89}}$
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3
Convert to a decimal to find the harmonic mean of your set of numbers. To convert a fraction to a decimal, divide the numerator by the denominator. ^{[5]
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Research source}
- For example:
  ${\frac {960}{89}}$
  $=960\div 89$
  $=10.79$
  So, the harmonic mean of the numbers 10, 12, 16, and 8 is 10.79.

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Method 3

Method 3 of 3:

Calculating the Harmonic Mean using Excel

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1
Enter the values into your spreadsheet. Make sure to only place one value in each cell.
- For example, if you need to find the harmonic mean of 10, 12, 16, and 8, you might type each of these values into a separate cell in the spreadsheet, cells A1-A4.
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2
Enter the function for the harmonic mean. The function is HARMEAN(number 1, [number 2]...). ^{[6]
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Research source} To select the function, begin typing “=HARMEAN” into a blank cell of the spreadsheet, then double-click on the function when it pops up.
- For example, type “=HARMEAN” into cell A5 of your spreadsheet and double-click on the function.
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3
Highlight the cells containing the values you are averaging. Hit the enter key. Excel will calculate the harmonic mean for you and display it in the spreadsheet.
- For example, highlight cells A1-A4 of your spreadsheet and hit enter. Excel will calculate 10.78652 as the harmonic mean.

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Question

What are the harmonic means of 5, 6 and 7?

Community Answer

3/(1/5+1/6+1/7) = 3/(42/210 + 35/210 + 30/210 ) = 3/( 107/210) = 3 x 210/107= 630/107 = 5.89

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What is the harmonic mean of x1,x2,x3.xn?

Community Answer

This is just asking for the formula: n/(1/x1 + 1/x2 + 1/x3 +...+1/xn)

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Can you tell me the harmonic mean of 0 2 4 6?

Donagan

Top Answerer

There's no harmonic mean when zero is one of the values in the set. That's because the formula would have 1/0 as one of the terms in the denominator, and 1/0 is infinity and not usable.

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About This Article

Reviewed by:

Chemistry Instructor

This article was reviewed by Anne Schmidt . Anne Schmidt is a Chemistry Instructor in Wisconsin. Anne has been teaching high school chemistry for over 20 years and is passionate about providing accessible and educational chemistry content. She has over 9,000 subscribers to her educational chemistry YouTube channel. She has presented at the American Association of Chemistry Teachers (AATC) and was an Adjunct General Chemistry Instructor at Northeast Wisconsin Technical College. Anne was published in the Journal of Chemical Education as a Co-Author, has an article in ChemEdX, and has presented twice and was published with the AACT. Anne has a BS in Chemistry from the University of Wisconsin, Oshkosh, and an MA in Secondary Education and Teaching from Viterbo University. This article has been viewed 157,511 times.

27 votes - 77%

Co-authors: 15

Updated: January 30, 2023

Views: 157,511

Categories: Probability and Statistics

Article Summary X

To calculate the harmonic meaning, start by determining the number of values in your set of numbers. For example, if you're working with 10, 12, 16, and 8, you have 4 numbers, so the value is 4. Then, rewrite the numbers you're working with as denominators over the number 1. For example, if you're working with 10, 12, 16, and 8, write them as 1/10, 1/12, 1/16, and 1/8. Then, divide 4 by the sum of the fractions to find the harmonic mean. To calculate the harmonic meaning using a calculator, keep reading!

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