# Solving Word Problems Involving Chebyshev's Theorem

**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 I is at least 1 – 1/k^2.**

**Sample Problem Number One :**

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

**Solution : 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%**

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

**Sample Problem Number Two :**

**The mean age of 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 ?**

**Solution : First find the value of k**

**40 – 20 = 8k**

**20 = 8k**

**k = 2.5**

**To find the percentage : 1 – 1/(2.5)^2 = 84 %**

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

**Sample Problem Number Three :**

**The mean age of saleslady in ABC Dept Store is 30 with a standard deviation of 6 . Between which two age limit must 75% of the data set lie ?**

**Solution : 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**

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

**Sample Problem Number Four :**

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

**Solution : 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**

**The mean score of 60 with an sd of 10 must lie between 50 and 110 to represent 88.89% of the data set.**

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## Comments 4 comments

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.

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.

Thank you for the questions Ma'am