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## How To Use Chebyshev’s Theorem Calculator

You can use Chebyshev’s Theorem Calculator on any shaped distribution.  The calculator shows you the smallest percentage of data values in “k” standard deviations of the mean.  Then, you will get a step-by-step explanation on how to do it yourself.  You don’t need the mean and standard deviation to use this calculator.

You can use the Chebyshev’s Theorem Calculator as a learning tool.  The best approach is to first look at a sample solution to a couple different problems and understand the steps shown in the solution.  Then, try a problem on your own using the same strategy, then check your work with the calculator.  You GOT this!

## Chebyshev’s Theorem Explained

So what is Chebyshev’s Theorem in statistics and what is Chebyshev’s Theorem used for?  We use Chebyshev’s Theorem, or Chebyshev’s Rule, to estimate the percent of values in a distribution within a number of standard deviations.  That is, any distribution of any shape, whatsoever.  That means, we can use Chebyshev’s Rule on skewed right distributions, skewed left distributions, bimodal distributions, etc.  For that reason, the estimate is conservative. We use the words “at least” when describing the percentage of data values.   You will see the use of “at least” in the Chebyshev’s Theorem problems and answers given below.  You can use Chebyshev’s Theorem Calculator above to see solutions to any problem you may have.

If you know that the distribution you are working with is a bell-shaped distribution, and you want to find the percentage of data values within 1, or 2, or 3 standard deviations, then you can use the Empirical Rule Calculator, a bell-shaped distribution percentage calculator.  Teachers and textbooks often discuss Chebyshev’s Theorem and the Empirical Rule together.  It’s important to remember that we only use the Empirical Rule with bell-shaped distributions.

• Chebyshev’s Theorem / Chebyshev’s Rule – used for any shaped distribution
• Empirical Rule – used only for bell-shaped distributions

## Chebyshev’s Theorem Definition

Chebyshev’s Formula: percent of values within k standard deviations = $1 – \frac{1}{k^2}$

For any shaped distribution, at least $1 – \frac{1}{k^2}$ of the data values will be within k standard deviations of the mean.  The value for k must be greater than 1.  Using Chebyshev’s rule in statistics, we can estimate the percentage of data values that are 1.5 standard deviations away from the mean.  Or, we can estimate the percentage of data values that are 2.5 standard deviations away from the mean.  The Chebyshev’s Theorem calculator, above, will allow you to enter any value of k greater than 1.  The Chebyshev calculator will also show you a complete solution applying Chebyshev’s Theorem formula.

## Chebyshev’s Theorem Example Problems

We’ll now demonstrate how to apply Chebyshev’s formula with specific examples.  These Chebyshev’s Theorem practice problems should give you an understanding on using Chebyshev’s Theorem and how to interpret the result.

### Example 1

A distribution of student test scores is skewed left.  Using Chebyshev’s Rule, estimate the percent of student scores within 1.5 standard deviations of the mean.
Solution:
The value of k in this problem is 1.5, so we substitute in 1.5 in Chebyshev’s formula:
$$1 – \frac{1}{1.5^2}$$
Squaring the value of k, we have
$$k^2 = 1.5^2 = 2.25$$
Divide 1 by 2.25
$$\frac{1}{2.25} = 0.4444$$
Subtract 0.4444 from 1
$$1 – 0.4444 = 0.5556$$
Multiply by 100 to convert the answer into a percent
$$0.5556 \cdot 100 = 55.56%$$Interpretation:
At least 55.56% of the test scores in the skewed left distribution are within 1.5 standard deviations of the mean.  That is, from 1.5 standard deviation below to 1.5 standard deviations above the mean.

### Example 2

A distribution of student credit scores is skewed right.  Using Chebyshev’s Rule, estimate the percent of credit scores within 2.5 standard deviations of the mean.
Solution:
The value of k in this problem is 2.5, so we substitute in 2.5 in Chebyshev’s formula:
$$1 – \frac{1}{2.5^2}$$
Squaring the value of k, we have
$$k^2 = 2.5^2 = 6.25$$
Divide 1 by 6.25
$$\frac{1}{6.25} = 0.16$$
Subtract 0.16 from 1
$$1 – 0.16 = 0.84$$
Multiply by 100 to convert the answer into a percent
$$0.84 \cdot 100 = 84%$$Interpretation:
At least 84% of the credit scores in the skewed right distribution are within 2.5 standard deviations of the mean.  That is, from 2.5 standard deviation below to 2.5 standard deviations above the mean.

You can enter 1.5 and 2.5 into the Chebyshev’s Theorem Calculator above an verify the same results shown here.

## Chebyshev’s Theorem in Excel

An alternative to using Chebyshev’s Theorem Calculator is to calculate Chebyshev’s Theorem in Excel.  You can download the linked spreadsheet here.

The spreadsheet linked here contains the formulas needed to apply Chebyshev’s Theorem in Excel.  The parts of the spreadsheet are explained in the 3 steps below.

1. In cell A2, enter the number of standard deviations.  As an example, I am using 1.2 standard deviations.
2. In cell B2, enter the Chebyshev Formula as an excel formula. In the formula, multiply by 100 to convert the value into a percent:

=(1-1/A2^2)*100.  Use cell A2 to refer to the number of standard deviations.

3. Press Enter, and get the answer in cell B2.  Round to the nearest hundredth, and the answer is 30.56%.

Interpretation: At least 30.56% of the data values are within 1.2 standard deviations of the mean.

## Chebyshev’s Theorem Practice Problems Given the Mean and Standard Deviation

There is no need for knowing the mean or standard deviation to use Chebyshev’s Rule, but if the problem provides these values, you can interpret the result further.  For example, let’s take a look at the examples from above, but this time let’s suppose the problems provide a mean and a standard deviation.

### Example 1

A distribution of student test scores is skewed left.  Using Chebyshev’s Rule, estimate the percent of student scores within 1.5 standard deviations of the mean.  Mean = 70, standard deviation = 10.
Solution:
Using Chebyshev’s formula by hand or Chebyshev’s Theorem Calculator above, we found the solution to this problem to be 55.56%.  Now, let’s incorporate the given mean and standard deviation into the interpretation.
First, calculate 1.5 standard deviations.
$$\text{1.5 standard deviations} = 1.5 \cdot 10$$
$$\text{1.5 standard deviations} = 15$$
Next, subtract and add 1.5 standard deviations from/to the mean, 70.
$$70 – 15 = 55$$
$$70 + 15 = 85$$
So now, the interpretation of the problem becomes: At least 55.56% of the test scores in the skewed left distribution are between 55 and 85.

### Example 2

A distribution of student credit scores is skewed right.  Using Chebyshev’s Rule, estimate the percent of credit scores within 2.5 standard deviations of the mean.  Mean = 400, standard deviation = 120.
Solution:
UsingChebyshev’s formula by hand or Chebyshev’s Theorem Calculator above, we found the solution to this problem to be 84%.  Now, let’s incorporate the given mean and standard deviation into the interpretation.
First, calculate 2.5 standard deviations.
$$\text{2.5 standard deviations} = 2.5 \cdot 120$$
$$\text{2.5 standard deviations} = 300$$
Next, subtract and add 2.5 standard deviations from/to the mean, 400.
$$400 – 300 = 100$$
$$400 + 300 = 700$$
So now, the interpretation of the problem becomes: At least 84% of the credit scores in the skewed right distribution are between 100 and 700.

## More Challenging Practice Problems Verified with the Chebyshev’s Theorem Calculator

Here are some challenging practice problems that use Chebyshev’s Theorem.

### Example 1

A distribution of 800 student test scores is skewed left.  Using Chebyshev’s Rule, what’s the smallest number of scores within 2 standard deviations of the mean?

Solution:
To solve this problem, we’ll apply Chebyshev’s Rule as we did in the examples above, and then we’ll take one more step.  We’ll multiple the percentage we find by 800, the size of the distribution.
You can either use the Chebyshev’s Theorem Calculator above to find the percentage, or calculate the percentage by hand using the formula.
The value of k in this problem is 2, so we substitute in 2 in Chebyshev’s formula:
$$1 – \frac{1}{2^2}$$
Squaring the value of k, we have
$$k^2 = 2^2 = 4$$
Divide 1 by 4
$$\frac{1}{4} = 0.25$$
Subtract 0.25 from 1
$$1 – 0.25 = 0.75$$
Now, multiply by 800 to get 75% of 800.  This is the smallest number of student scores within 2 standard deviations of the mean.
$$0.75 \cdot 800 = 600$$

Interpretation:
At least 600 student scores in the skewed left distribution are within 2 standard deviations of the mean.  That is, from 2 standard deviation below to 2 standard deviations above the mean.

### Example 2

Using Chebyshev’s Rule, about 81.1% of data values are within ‘k’ standard deviations of the mean.  How many standard deviations is that?

Solution:
For this problem, we’ll need to solve for k in Chebyshev’s formula.  Begin by setting the formula equal to .8110, the percentage expressed as a decimal.
$$.8110 = 1 – \frac{1}{k^2}$$
Add $\frac{1}{k^2}$ to both sides
$$.8110 + \frac{1}{k^2}= 1$$
Subtract .8110 from both sides
$$\frac{1}{k^2} = 1 – .8110$$
Evalulate the expression on the right
$$\frac{1}{k^2} = .189$$
Multiply both sides by $k^2$.
$$1 = .189 \cdot {k^2}$$
Divide both sides by .189
$$5.29100529101 = k^2$$
Take the square root of 5.29100529101
$$\sqrt{5.29100529101} = k$$
$$k = 2.3$$

Interpretation: According to Chebyshev’s Theorem at least 81.1% of the data values in the distribution are within 2.3 standard deviations of the mean.
You can verify that 2.3 is the correct answer by inputing it into Chebyshev’s Theorem Calculator above.  You should get an answer of about .8110.

## Chebyshev’s Theorem Quiz

Test your knowledge of Chebyshev's Theorem with this online quiz.  You can first try to do the work by hand, then check your work with the Chebyshev's Theorem Calculator.

## General Notes

Questions involving Chebyshev’s Theorem in introductory statistics classes are usually not too difficult.  Key phrases to look out for are the type of distribution.  If you see the phrase “bell-shaped distribution” or “normal distribution,” then you should not be using Chebyshev’s rule to estimate percentages in the distribution.  Therefore, use the Empirical Rule in that case instead.  Questions that require the use of Chebyshev’s Rule will note that the distribution is non-bell-shaped, skewed right, skewed left, bimodal, j-shaped, etc.  Knowing the type of distribution will guide you in how to solve the problem, either Chebyshev’s Theorem or the Empirical Rule.

## Next Steps

Now that you’ve learned all about Chebyshev’s Theorem, go and have a look at the Empirical Rule Calculator.  The empirical rule only works with bell-shaped distributions, but the estimates are more precise than with Chebyshev’s Rule.