Statistics Chebyshev's Theorem - Statistics

What is Chebyshev's Theorem?

The fraction of any set of numbers include between k standard deviations of same numbers of the mean of those numbers should be
k standard deviations
Where −
k standard deviations
and kk must be greater than 1

Example

Problem Statement:

Here we use Chebyshev's theorem to know the percent of the values will include between 123 and 179 for a data set with mean of 151 and standard deviation of 14.

Solution:

  • When we subtract 151-123 and get 28, which meant to be that 123 is 28 units below the mean.
  • When we subtract 179-151 and also get 28, which meant to be that 151 is 28 units above the mean.
  • So the values together explain us that values between 123 and 179 are all within 28 units of the mean. Therefore the "within number" is 28.
  • If you assume that the number of standard deviations, k, which the "within number", 28, amounts to by dividing it by the standard deviation:
k standard deviations
Let’s assume that the values between 123 and 179 lying with 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:
k standard deviations
So 3/4 of the data may include between 123 and 179. And since 3/4=75% that implies that 75% of the data values are between 123 and 179.

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