Summary: This module explains Chevyshev’s Theorem as it pertains to the spread of non-normal data. Given an data set, Chevyshev’s Theorem gives a worst case scenario for the percentage of data within a given number of standard deviations from the mean.

*Chevyshev’s Theorem*

The proportion (or fraction) of any data set lying within K standard deviations of the mean is always at least 1 -

For K = 2, the proportion is 1 -

For K = 3, the proportion is 1 -

Using the data from the pre-calculus class exams and K = 2, this means that at least 75% of the scores fall between 73.5 - 2(17.9) and 73.5 + 2(17.9), or between 37.7 and 109.3.

In actual fact all but one data value falls in this range, however Chevyshev's Theorem gives the worst case scenario.

Using the pre-calculus class exams, what would the range of values be for at least 89% of the data according to Chevyshev’s Theorem?

73.5 – 3(17.9) to 73.5 + 3(17.9) or 19.8 to 127.2

Using Chevyshev’s Theorem, what percent of the data would fall between 46.65 and 100.35?

Step 1: Find how far the maximum (or minimum) value is from the mean. 100.35 – 73.5 = 26.85

Step 2: How many standard deviations does 26.85 represent? 26.85/17.9 = 1.5. Hence K = 1.5

Step 3: If K = 1.5, then the percentage is

Given a data set with a mean of 56.3 and a standard deviation of 8.2, use this information and Chevyshev’s Theorem to answer the following questions.

What percent of the data lies within 2.2 standard deviation from the mean?

For the given sent of data, about 79% of the data falls between which two values?

56.3 – 2.2(8.2) = 38.26 and 56.3 + 2.2(8.2) = 74.34

What percent of the data lies between the values 45.64 and 66.96?

Step 1: Find how far the maximum value is from the mean: 66.96 – 56.3 = 10.66

Step 2: How many standard deviations does 10.66 represent? 10.66/8.2 = 1.3. Hence K = 1.3

Step 3: If K = 1.3, then the percentage is 1 -