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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 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 size 12{ { {1} over {2 rSup { size 8{2} } } } } {} = 1 - 1 4 size 12{ { {1} over {4} } } {} = 3 4 size 12{ { {3} over {4} } } {} , hence 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 size 12{ { {1} over {3 rSup { size 8{2} } } } } {} = 1 - 1 9 size 12{ { {1} over {9} } } {} = 8 9 size 12{ { {8} over {9} } } {} , hence 8 9 size 12{ { {8} over {9} } } {} ths or approximately 89% of the data falls within 3 standard deviations of the mean.

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 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.

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

1 - 1 2.2 2 0.793 or 79%

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

Questions & Answers

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The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Collaborative statistics (with edits: teegarden). OpenStax CNX. Jul 20, 2009 Download for free at http://legacy.cnx.org/content/col10561/1.3
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