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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 −
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${k = frac{the within number}{the standard deviation}}$
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 −
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We subtract 151-123 and get 28, which tells us that 123 is 28 units below the mean.
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We subtract 179-151 and also get 28, which tells us that 151 is 28 units above the mean.
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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.
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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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