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.

**Contents:**

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 / k^{2} )), 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.

**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:

2^{2} = **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.

That’s:

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

.

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.

*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*.

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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 ?

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.

Can you please help me to understand well chebyschev’s theorem?

What is it you want to know?

How about calculating approximately the proportion of observations are less than -1.27???

I’m not sure what you are trying to do. I *think* you might be referring to a z-score of -1.27? If so, see: one tailed distribution for the calculation steps.