# 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 Maria Cristina Aquino Santander**

## Comments

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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