# Solving Word Problems Involving Chebyshev's Theorem

Chebyshev’s theorem states that the proportion or percentage of any data set that lies within *k* standard deviation of the mean where *k* is any positive integer greater than *1* is at least *1 – 1/k^2*.

Below are four sample problems showing how to use Chebyshev's theorem to solve word problems.

## Sample Problem One

The mean score of an Insurance Commission Licensure Examination is 75, with a standard deviation of 5. What percentage of the data set lies between 50 and 100?

First find the value of *k*.

*Mean – (k) (sd) = lower limit*

*75 – 5K - 50*

*75 – 50 = 5k*

*25 = 5k*

*K = 5*

To get the percentage use 1 – 1/k^2.

*1 - 1/ 25 = 24/25 = 96%*

**Solution:** 96% of the data set lies between 50 and 100.

## Sample Problem Two

The mean age of a flight attendant of PAL is 40 years old, with a standard deviation of 8. What percent of the data set lies between 20 and 60?

First find the value of *k.*

*40 – 20 = 8k*

*20 = 8k*

*k = 2.5*

Find the percentage.

*1 – 1/(2.5)^2 = 84%*

**Solution:** 84% of the data set lies between the ages 20 and 60.

## Sample Problem Three

The mean age of salesladies in an ABC department store is 30, with a standard deviation of 6 . Between which two age limits must 75% of the data set lie?

First find the value of *k.*

*1 - 1/k^2 = ¾*

*1 - ¾ = 1/k^2*

*¼ = 1/k^2*

*k^2 = 4*

*k = 2*

Lower age limit:

*30 - (k ) (sd) = 30 - (6)(2) = 30 -12 = 18*

Upper age limit:

*30 + ( k) (sd) = 30 + (6)(2) = 30 + 12 = 42*

**Solution:** The mean age of 30 with an standard deviation of 6 must lie between 18 and 42 to represent 75% of the data set.

## Sample Problem Four

The mean score on an accounting test is 80, with a standard deviation of 10. Between which two scores must this mean lie to represent 8/9 of the data set?

Find first the value of *k.*

*1 - 1 /k^2 = 8/9*

*1 - 8/9 = 1/k^2*

*1/9 = 1/k^2*

*k^2 = 9*

*k = 3*

Lower limit:

*80 – (10)(3) = 80 – 30 = 50*

Upper limit:

*80 + 30 = 110*

**Solution:** The mean score of 60 with a standard deviation of 10 must lie between 50 and 110 to represent 88.89% of the data set.

**© 2012 Cristine Abigail **

## Comments

Question 4

The administrator of a hospital surveyed the number of days.

200 randomly chosen patients stayed in the hospital following an operation. The data are given below:

Hospital Stay in Days 1-3 4-6 7-9 10-12 13-15 16-18 19-21 22-24

Frequency 18 90 44 21 9 9 4 5

a) Calculate Mean, Median and Mode of the data and Compare the results?

b) Calculate the standard deviation and coefficient of variation.

c) According to Chebyshev’s theorem, how many stays should be between 0 and 17 days? How many are actually in that interval?

d) Because the distribution is roughly bell-shaped, how many stays can we expect between 0 and 17 days?

In sample 3 how did you get 1/4 to 4 I missed that step. It's been over 40 years since I've done these problems and I forget

Thank you

Nice and simple examples

Fantastic lesson

Thankyou for uploading it. Very useful and clear all my doubts regarding this theorem

Thanks for appreciating this solution

i like it

Thanks for dropping by. Stay blessed.

this is very usefull for me

thanks author of this site

Wow, to be frank this by far the best site for chebyshev's theory explanation. Examples are on point and clear. I really love this site

Thank you for the questions Ma'am

I like math though I'm not good at it. I use stat especially when I do my Training Needs analysis. I need to back up my training proposals with hard facts. Statistics often gets the training approved.

This brings back painful memories. But it's nice to review numbers once again. I just don't know why I took an extra elective in math class in 3rd year high school. I'm a certified nerd!

But seriously, I use SPSS for any staistical computation I need for work. Of course doing it the old fashioned manual computation is important.

Thanks for the clear discussion. I need to bookmark this hub.

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