Chebyshev’s Theorem shows you how to use the mean and standard deviation to find the percentage of total observations that fall within a given interval around the mean.

For every number k greater than 1 at least (1 –

1

k^{2}

) of the data values are k standard deviations of the mean.

## Important things to keep in mind about Chebyshev’s theorem

Look at the following illustration and make the following observations:

The distance between µ and µ + kσ is kσ, since µ + kσ – µ = kσ

The distance between µ and µ – kσ is kσ, since µ – (µ – kσ) = µ – µ – – kσ = 0 – – kσ = kσ

The answer to (1 –

1

k^{2}

) is usually expressed as a percentage after you finish calculating.

Notice that we are saying that k is greater than 1 or at least 2. k must be 2 because we get 0 when k = 1.

Here is what we get when k = 1

= 1 – 1 = 0

## How to find the standard deviation using Chebyshev’s theorem

Now let’s see how we can apply Chebyshev’s theorem.

For example, suppose you want to find the percentage of values in a data set that are within 2 standard deviations of the mean.

Just replace k = 2 in the formula.

= 1 – 0.25 = 0.75

0.75 as percent is 75%

Therefore, 75% of the values in a data set are within 2 standard deviations of the mean.

How do we use the mean and standard deviation to find the percentage of total observations that fall within a given interval around the mean?

Assuming µ = 39 and σ = 5, find the percentage of values that are between 29 and 49 of the mean.

29 ——– 39 ——————– 49

We just need to find k and there are 2 ways to do it.

First notice, as mentioned earlier, that the distance between 39 and 49 is equal to kσ

Since the distance between 39 and 49 is 10, kσ = 10

kσ = 10

σ = 5, so 5k = 10

Since 5 times 2 = 10, k = 2.

We calculated the percentage earlier when k = 2. We found it to be 75%.

Therefore, 75% of the values are between 29 and 49 of the mean.

The **second** To find the answer, you need to find that µ + kσ = 49

29 —————– 39 —————– 49

µ – kσ µ µ + kσ

µ + kσ = 49

39 + kσ = 49

39-39 + kσ = 49-39

0 + kσ = 10

kσ = 10

5k = 10 and k = 2