Statistics - Chebyshev's Theorem

The fraction of any set of numbers lying within k standard deviations of those numbers of the mean of those numbers is at least

${1-\frac{1}{k^2}}$

Where −

${k = \frac{the\ within\ number}{the\ standard\ deviation}}$

and ${k}$ must be greater than 1

Example

Problem Statement −

Use Chebyshev's theorem to find what percent of the values will fall between 123 and 179 for a data set with mean of 151 and standard deviation of 14.

Solution −

We subtract 151-123 and get 28, which tells us that 123 is 28 units below the mean.
We subtract 179-151 and also get 28, which tells us that 151 is 28 units above the mean.
Those two together tell us that the values between 123 and 179 are all within 28 units of the mean. Therefore the "within number" is 28.
So we find the number of standard deviations, k, which the "within number", 28, amounts to by dividing it by the standard deviation −

${k = \frac{the\ within\ number}{the\ standard\ deviation} = \frac{28}{14} = 2}$

So now we know that the values between 123 and 179 are all within 28 units of the mean, which is the same as within k=2 standard deviations of the mean. Now, since k > 1 we can use Chebyshev's formula to find the fraction of the data that are within k=2 standard deviations of the mean. Substituting k=2 we have −

${1-\frac{1}{k^2} = 1-\frac{1}{2^2} = 1-\frac{1}{4} = \frac{3}{4}}$

So ${\frac{3}{4}}$ of the data lie between 123 and 179. And since ${\frac{3}{4} = 75}$% that implies that 75% of the data values are between 123 and 179.

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