Connexions module: m29218 1
Chevyshev's Theorem
∗
Mary Teegarden
This work is produced by The Connexions Project and licensed under the
Creative Commons Attribution License
†
Abstract
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
K2
, where K is any positive number greater then 1. Why is this?
For K = 2, the proportion is 1 -
1
22
=1-
1
4
=
3
4
, hence
3
4
ths or 75% of the data falls within 2 standard
deviations of the mean.
For K = 3, the proportion is 1 -
1
32
= 1 -
1
9
=
8
9
, hence
8
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
(Solution on p. 3.)
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?
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
1.52≈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
(Solution on p. 3.)
What percent of the data lies within 2.2 standard deviation from the mean?
Exercise 3
(Solution on p. 3.)
For the given sent of data, about 79% of the data falls between which two values?
∗
Version 1.4: Jul 15, 2009 6:23 pm GMT-5
†
http://creativecommons.org/licenses/by/3.0/
http://cnx.org/content/m29218/1.4/
