1. Home
  2. Collaborative statistics (with
  3. Descriptive statistics
  4. Chevyshev’s theorem

2.9 Chevyshev’s theorem

<< Chapter < Page
  Collaborative statistics (with   Page 1 / 1
Chapter >> Page >
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

how did the oxygen help a human being
Achol Reply
how did the nutrition help the plants
Achol Reply
Biology is a branch of Natural science which deals/About living Organism.
Ahmedin Reply
what is phylogeny
Odigie Reply
evolutionary history and relationship of an organism or group of organisms
AI-Robot
ok
Deng
what is biology
Hajah Reply
cell is the smallest unit of the humanity biologically
Abraham
what is biology
Victoria Reply
what is biology
Abraham
HOW CAN MAN ORGAN FUNCTION
Alfred Reply
the diagram of the digestive system
Assiatu Reply
allimentary cannel
Ogenrwot
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Collaborative statistics (with edits: teegarden). OpenStax CNX. Jul 20, 2009 Download for free at http://legacy.cnx.org/content/col10561/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Collaborative statistics (with edits: teegarden)' conversation and receive update notifications?

Ask