Statistics How To

Chebyshev’s Theorem: How to Calculate it by Hand and in Excel

Hypothesis Testing > Chebyshev’s Theorem

Before you start, you may want to read this article on Chebyschev’s Inequality, which explains when you would want to use the theorem. Chebyshev’s Interval refers to the intervals you want to find when using Chebyshev’s theorem.

How to calculate Chebyshev’s theorem by hand.
How to calculate Chebyshev’s Theorem in Excel.
Where did Chebyshev’s Theorem come from?

How to Calculate Chebyshev’s Theorem

Watch the video or read the steps below:

For normal distributions, about 68% of results will fall between +1 and -1 standard deviations from the mean. About 95% will fall between +2 and -2 standard deviations. Chebyshev’s Theorem allows you to use this idea for any distribution, even if that distribution isn’t normal. The theorem states that for a population or sample, the proportion of observations is no less than (1 – (1 / k2 )), as long as the z score’s absolute value is less than or equal to k. You can only use Chebyshev’s Theorem to get results for standard deviations more than 1; It can’t be used to find results for smaller values like 0.1 or 0.9. Technically, you could use it and get some kind of a result, but those results wouldn’t be valid.

Chebyshev's Theorem

In a normal distribution, the percentages of scores you can expect to find for any standard deviations from the mean are the same.

Sample problem: a left-skewed distribution has a mean of 4.99 and a standard deviation of 3.13. Use Chebyshev’s Theorem to find the proportion of observations you would expect to find within two standard deviations from the mean:

Step 1: Square the number of standard deviations:
22 = 4.

Step 2: Divide 1 by your answer to Step 1:
1 / 4 = 0.25.

Step 3: Subtract Step 2 from 1:
1 – 0.25 = 0.75.

At least 75% of the observations fall between -2 and +2 standard deviations from the mean.
mean – 2 standard deviations
4.99 – 3.13(2) = -1.27
mean + 2 standard deviations
4.99 + 3.13(2) = 11.25
Or between -1.27 and 11.25

That’s it!


Warning: As you may be able to tell, the mean of your distribution has no effect of Chebyshev’s theorem! That fact can cause some wide variations in data, and some inaccurate results.

Like the explanation? Check out the Practically Cheating Statistics Handbook, which has hundreds more step-by-step explanations, just like this one!

How to Calculate Chebyshev’s Formula in Excel.

Microsoft Excel has a wide variety of built-in functions and formulas that can help you with statistics. However, it does not have a built-in formula for Chebyshev’s Theorem. In order to calculate Chebyshev’s theorem in Excel, you’ll need to add the formula yourself. If you want to use Chebyshev’s formula just once or twice, you can type the formula into a cell. However, if you intend on using Chebyshev’s formula several times over time, you can add a custom function (=CHEBYSHEV) to Microsoft Excel.

How to Calculate Chebyshev’s Formula in Excel (Temporary Use).

Step 1: Type the following formula into cell A1: =1-(1/b1^2).
Step 2: Type the number of standard deviations you want to evaluate in cell B1.
Step 3: Press “Enter.” Excel will return the percentage of results you can expect to find within that number of standard deviations in cell B1.

How to Calculate Chebyshev’s Formula in Excel (Adding a Custom Formula)

Step 1: Open the Visual Basic editor in Excel. To open the Visual Basic editor, click the “Developer” tab and then click “Visual Basic.”

Step 2: Click “Insert” and then click “New Module.”

Step 3: Type the following code into the blank window:

Function Chebyshev(stddev)
If stddev >= 0 Then
Chebyshev = (1 – (1 / stddev ^ 2))
Else: Chebyshev = 0
End If
End Function

Typing the code for a Chebyschev custom function in Microsoft Excel.

Typing the code for a Chebyshev custom function in Microsoft Excel.

Step 4: Close the visual basic window and return to the worksheet. The custom function is now ready to use: type “=chebyshev(x)” into a blank cell, where “x” is the number of standard deviations. Excel will calculate Chebyshev’s theorem and return the result in the same cell.

Where did Chebyshev’s Theorem Come From?

Pafutny Lvovich Chebyshev (1821-1894)was a Russian mathematician. His friend, mathematician and gifted linguist Irenée-Jules Bienaymé translated many of Chebyshev’s works into French.

In 1867 Chebyshev published a paper On mean values which first mentioned the inequality to give a generalized law of large numbers. however, the inequality actually first appeared fourteen years earlier in Bienaymés Considérations à l’appui de la découverte de Laplace. The editor who discovered Chebyshev’s use of Bienaymé’s inequality (without mention of the original author) said:

It is a pity that their common interest in the Inequality somehow “slipped through the cracks” in the early contacts between Bienaymé and Chebyshev. Possibly the Inequality was regarded by Bienaymé as a minor result compared with his main themes of linear least squares and Laplacian defence. Chebyshev’s recognition of its significance and its clear statement has, at any rate, always been a defensive point in his favour stressed by some historiographers. From The University of St. Andrews.

Chebyshev’s theorem is often spelled many different ways — you’ll find it spelled as Chebychev’s theorem, Chebyschev’s theorem and even Tschebyscheff’s theorem. That’s mostly due to the fact that his original name was Russian, which uses a different alphabet (cyrillic). “Chebyshev” is just the word, taken as it sounds and translated into an English approximation.

Fun fact: There is a crater on the moon named after him: Crater Chebyschev.

Chebyshev's crater on the moon.

Chebyschev’s crater on the moon.

More info: Click here for a full explanation of Chebyshev’s Inequality.

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you’re are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

Chebyshev’s Theorem: How to Calculate it by Hand and in Excel was last modified: September 2nd, 2017 by Stephanie

18 thoughts on “Chebyshev’s Theorem: How to Calculate it by Hand and in Excel

  1. Lyon

    what do you mean by “z” here in this sentence–as long as the z score’s absolute value is less than or equal to k? thank you.

  2. Andale

    I’m working on this year. is also in the works :)

  3. Andale

    K is the number of standard deviations either side of the mean. You shouldn’t need to find it — it’s usually stated in the question/problem you’re given (in the sample problem, you’re finding how many points fall within two standard deviations, so k=2).


  4. Sammy

    Use Chebyshev’s theorem to find what percent of the values will fall between 236 and 338 for a data set with a mean of 287 and standard deviation of 17.

  5. Nasim

    dear all the above question is still not clear for me, becuase it only used the formula not that mean and standard diviation as he was exampled, please anyone clear that for me

  6. rube

    what if the question is a little different like avg car in the lot = 200,000 and the standard deviation is 32,000 and the deviation but the question is something like find the range in which 80% of of cars fall?? I can’t see how to fit this formula to that. Thanks a lot!

  7. Shahrokh

    Hi dear admin and friends.
    I have to design a 4 linkage grashove system and I should use Chebyshev’s Theorem for calculating the bars’ length. please help me ASAP… I really don’t know which degrees I should choose for my function. y=tanx

  8. Chris

    Uhmm .. excuse me what is then the use of the given mean and standard deviation if they are not needed in the equation and does it mean that k=2 will always be 75%? even if the distribution is different ?

  9. Andale

    Yes, k=2 will always be 75%.
    The mean and standard deviation aren’t used in the equation, but you’ll need them to figure out the spread. I added clarification for this under the last step.

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