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Chevyshev’s Theorem

Module by: Mary Teegarden. E-mail the author

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 - 1 K 2 1 K 2 size 12{ { {1} over {K rSup { size 8{2} } } } } {} , where K is any positive number greater then 1. Why is this?

For K = 2, the proportion is 1 - 1 2 2 1 2 2 size 12{ { {1} over {2 rSup { size 8{2} } } } } {} = 1 - 1 4 1 4 size 12{ { {1} over {4} } } {} = 3 4 3 4 size 12{ { {3} over {4} } } {} , hence 3 4 3 4 size 12{ { {3} over {4} } } {} ths or 75% of the data falls within 2 standard deviations of the mean.

For K = 3, the proportion is 1 - 1 3 2 1 3 2 size 12{ { {1} over {3 rSup { size 8{2} } } } } {} = 1 - 1 9 1 9 size 12{ { {1} over {9} } } {} = 8 9 8 9 size 12{ { {8} over {9} } } {} , hence 8 9 8 9 size 12{ { {8} over {9} } } {} ths or approximately 89% of the data falls within 3 standard deviations of the mean.

Example 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.

Exercise 1

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?

Solution

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

Example 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 1 - 1 1.5 2 0.55556 1- 1 1.5 2 0.55556 , or approximately 56%

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.

Exercise 2

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

Solution

1 - 1 2.2 2 0.793 1- 1 2.2 2 0.793 or 79%

Exercise 3

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

Solution

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

Exercise 4

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

Solution

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 - 1 1 . 3 2 1 1 . 3 2 size 12{ { {1} over {1 "." 3 rSup { size 8{2} } } } } {} .408 or approximately 41%

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