Chebyshev’s Inequality

Describing any set of data

Lesson Overview

Goals

Students will be able to apply Chebyshev's Inequality to determine the lower bounds for several key values of k

Students will be able to determine the minimum percentage of data elements that lie within k standard deviations of any data set's mean

Instructor Presentation

Introduction

For any data set, regardless of the shape of the distribution, at least 100*(1-(1/(k^2))) of the data elements will lie within k standard deviations of the mean, where k is any number greater than 1.  That is, at least 100*(1-(1/(k^2)))  of the data will lie between   μ - kσ  and μ + kσ for k>1.

Lets look at several values for k and see just what the lower boundry for the percentage of observations are.  Plugging k into the formula 100*(1-(1/(k^2)))

graph=Table[{k,N[100*(1-(1/(k^2))),3]},{k,2,7}];
TableForm[Prepend[graph,{"k","Minimum %"}]]

From the above table, we can see that for any data set (skewed right, non-skewed, skewed left) the lowest percentage of observations that lie within 5 standard deviations of the mean would be 96%.    What percentage of values falls within 3 standard deviations?   Correct, 88.9%.

Concept Exploration

Interactive Demonstration

Move the Number of Std Dev (k) slider to change the values for k in the equation, 100 (1-(1/(k^2))).  This equation gives us the percentage of the data elements that fall within the range  μ-k σ  and μ + k σ.  Please note:

Observation 1:  That for any value of k, the resultant percentage remains constant.  This means that we can be assured that 100*(1-(1/(k^2))) percent of any data set falls between  μ - k σ  and μ + k σ.

Observation 2:  The values for larger values of k appear to get closer to 1.  Do they ever equal one?  Why?

Use the Zoom slider to get a better look at the values of Chebyshev's Inequality as k gets larger.

graph=Manipulate[ListPlot[Table[{x,N[100*(1-(1/(x^2))),6]},{x,2,m}],Joined->True,Filling->Top,PlotRange->{{0,50},{y,100}}],{{m,2,"Number of Std Dev (k)"},2,50,1},{{y,40,"Zoom"},40,98,1}]

Examples

Example 1

Assume k=4, calculate the interval and minimum percentage of data observations that within lie within the interval.

Chebychev's Inequality   states     100*(1-(1/(k^2))) percent of the data value for any data set will be contained within the range of  μ-k σ  and μ+k σ.

Determining the range is really straight forward at this point.  Substituting k=4 into the ranges, we obtain μ-4σ  and μ+4σ as the range.

Employing Mathematica's computational skills, we can quickly create a table of k values and the appropriate percentage of inclusive data.

chebyshevlist=Table[{k,N[100*(1-(1/(k^2))),4]},{k,2,10}];
TableForm[Prepend[chebyshevlist,{"k","Minimum %"}]]

Looking within the table, we see that for k= 4, we can conclude that given any data set, that at least 93.75% of all of the data observations will fall within the range  of   μ - 4σ  and μ + 4σ.

Example 2

You are given a bag full of numbered balls.  You are told that the expected value of the numbers on the balls is 55  and that the variance of numbered balls is 7.5.  You reach in the bag and pull out a ball numbered 23.   Should you be suprised?

We know that the standard deviation (σ) = (Sqrt[Variance])   =  (Sqrt[7.5])So  we want to know how many (k)  standard deviations away from the mean that the number 23 is.  Substituting values into μ - k σ = 23.  We get   55  - k ((Sqrt[7.5)]) =  23.    So  k  =  ((55  - 23)/(Sqrt[7.5]))   or 11.7.
So according to Chebyshev's Inequality, what percent of data value would fall inside the range  μ ± kσ?   So we substitute k = 11.7 into Chebyshev's Inequality    100 * (1-(1/(k^2)))   ->   100 * (1-(1/(11.7^2)))     ->  99.2695.    Chebyshev's Inequality states that at least 99.2695% of the balls will have a number greater than 23  but smaller than 87.    So Yes  we would be suprised if we pulled out a ball numbered 23.

Examples Answer Key

Example 1

Solved Above.

Example 2

Solving Example 2

meannumberballs=55;
varnumberballs=7.5;
pullednumber=23;
chebyshevvalue=((meannumberballs-pullednumber)/(Sqrt[varnumberballs])) 
chebyshevpercent= 100 * (1-(1/(chebyshevvalue^2)))

Problems

Easy

(1)  Determine the minimum percentage of data values that fall within Chebyshev's interval given:

	1a. 		k  =  6

	1b.		k  =  3

	1c.		k  =  10

	1d.		k  =  5

(2)  What are the ranges for Chebyshev's inequality given the values of  k in the above problem (1), and given the mean of 115 and the standard deviation of 13.6?

	2a.

	2b.

	2c.

	2d.

Story Problems

(3)  You are given a bag full of numbered balls.You are told that the expected value of the numbers on the balls is 20 and that the variance of numbered balls is 3.6.  You reach in the bag and pull out a ball numbered 27.    Should you be suprised?

(4)  What number of standard deviations (k) does Chebyshev's Inequality predict 97.5% of the all data values will fall within?

Reference Material

Data and concepts presented above were adopted from Sullivan, Statistics: Informed Decisions Using Data, Third Edition, Prentice Hall, 2010.