## How to Calculate Chebyshev’s Theorem

For any normal distribution, about 68% of results will fall between +1 and -1 standard deviations from the mean, and about 95% will fall between +2 and -2 standard deviations. Chebyshev’s Inequality, sometimes called Chebyshev’s Theorem, allows you to extend this idea to any distribution: even if that distribution isn’t normal. The theorem states that for a population or sample, the proportion of observations is no less than (1 – (1 / k2 )), as long as the z score’s absolute value is less than or equal to k. You can only use Chebyshev’s Theorem to get results for standard deviations over 1.
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Sample problem: a left-skewed distribution has a mean of 4.99 and a standard deviation of 3.13. Use Chebyshev’s Theorem to calculate the proportion of observation you would expect to find within two standard deviations (in other words, between -2 and +2 standard deviations) from the mean:

Step 1: Square the number of standard deviations:
22 = 4

1 / 4 = 0.25

Step 3: Subtract Step 2 from 1:
1 – 0.25 = 0.75

At least 75% of the observations fall between -2 and +2 standard deviations from the mean.

That’s it! Like the explanation? Check out our statistics how-to book, with a how-to for every elementary statistics problem type.

Warning: As you may be able to tell, the mean of your distribution has no effect of Chebyshev’s theorem! That fact can cause some wide variations in data, and some inaccurate results.

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## 21 Responses to “How to Calculate Chebyshev’s Theorem”

1. ### Lyon said:

Oct 31, 11 at 1:24 pm

what do you mean by “z” here in this sentence–as long as the z score’s absolute value is less than or equal to k? thank you.

2. ### Stephanie said:

Nov 08, 11 at 8:28 am

Hello, Lyon,

The z score taken from the z table (see the z table on this site),

Stephanie

3. ### homework help said:

Nov 09, 11 at 10:46 am

homework help…

[...]Statistics How To» Blog Archive » How to Calculate Chebyshev’s Theorem[...]…

4. ### helping math make sense said:

Nov 09, 11 at 11:04 am

helping math make sense…

[...]Statistics How To» Blog Archive » How to Calculate Chebyshev’s Theorem[...]…

5. ### Faty DIeye said:

Feb 08, 12 at 11:08 am

This was very helpful thank you!

6. ### oriakhi osazemen said:

Jul 05, 12 at 10:43 am

this was realy helpful. Wish there was one on mathematics.

7. ### Andale said:

Jul 06, 12 at 4:27 pm

I’m working on algebrahowto.com this year. MathHowTo.com is also in the works :)

8. ### Stephanie said:

Aug 17, 12 at 3:46 am

What do you mean by K? how do you find K?

9. ### Andale said:

Aug 17, 12 at 7:42 am

K is the number of standard deviations either side of the mean. You shouldn’t need to find it — it’s usually stated in the question/problem you’re given (in the sample problem, you’re finding how many points fall within two standard deviations, so k=2).

Stephanie

10. ### Sammy said:

Aug 21, 12 at 2:28 pm

Use Chebyshev’s theorem to find what percent of the values will fall between 236 and 338 for a data set with a mean of 287 and standard deviation of 17.
Help

11. ### Andale said:

Aug 22, 12 at 7:25 am

Hi, Sammy,

Unfortunately I’m stretched too far to help with homework in the comments section. But please feel free to post in our forums, where one of our mods should be able to help :)

Stephanie

12. ### Nasim said:

Oct 18, 12 at 4:09 am

dear all the above question is still not clear for me, becuase it only used the formula not that mean and standard diviation as he was exampled, please anyone clear that for me
thanks

13. ### What is Chebyshev's Inequality? | said:

Oct 30, 12 at 11:51 am

[...] How to Calculate Chebyshev's Theorem | said: Apr 14, 12 at 2:47 pm [...]

14. ### What is Chebyshev's theorem? | said:

Oct 30, 12 at 3:12 pm

[...] to help you find out what percentage of the data is clustered around the mean. You can read how to calculate Chebyshev’s inequality in this article. Although Chebyshev’s theorem usually means Chebyschev’s inequality in elementary [...]

15. ### Chelsea said:

Dec 16, 12 at 8:11 pm

I dont understand this at all a little more help please

16. ### Andale said:

Dec 17, 12 at 10:35 am

Hello, Chelsea,

Which part wasn’t clear?

Regards,
Stephanie

17. ### rube said:

Jan 04, 13 at 11:19 am

what if the question is a little different like avg car in the lot = 200,000 and the standard deviation is 32,000 and the deviation but the question is something like find the range in which 80% of of cars fall?? I can’t see how to fit this formula to that. Thanks a lot!

18. ### Shahrokh said:

Jan 11, 13 at 7:45 am

I have to design a 4 linkage grashove system and I should use Chebyshev’s Theorem for calculating the bars’ length. please help me ASAP… I really don’t know which degrees I should choose for my function. y=tanx

19. ### mooha said:

Apr 03, 13 at 10:58 am

how to i calculate the percentile?

20. ### Andale said:

Apr 07, 13 at 11:14 am

Hi, Mooha,
Can you post your question on our forum? Click the “forums” tab at the top. One of our mods will be happy to help :)
Stephanie

21. ### Andale said:

Apr 07, 13 at 11:22 am

Mooha,
Can you post your question in our forums? Click the “forums” tab at the top of the page. One of our mods will be happy to help,
Stephanie