Chebyshev's Theorem

Chebyshev's Theorem

The proportion of any data set within k standard deviations of the mean is always at least where k>1.

When k =2 and k = 3, we get the following specific results.

- at least 3/4 ( or 75%) of the data is within 2 standard deviations of the mean.

Note that 3/4 comes from = =

- at least 8/9 ( or about 89%) of the data is within 3 standard deviations of the mean.

Note that 8/9 comes from = =

Example

On a statistics test the mean was 76 and the standard deviation was was 11. What does Chebyshev's Theorem say about the percentage of test takers who have scores between 54 and 98?

Solution .

We first have to know how many standard deviations 98 is above the mean. Since the mean is 76 and the standard deviation is 11, we must solve 98 = 76 + 11k for k. The solution is 2. Note that 54 is two standard deviations ( or 22) below the mean of 76. Since the data range given in the problem corresponds to being within 2 standard deviations of the mean, Chebyshev's Theorem says that at least 75 % of the scores are between 54 and 98.