Chebyshev’s Theorem shows you how to use the mean and standard deviation to find the percentage of total observations that fall within a given interval around the mean.
For every number k greater than 1 at least (1 –
1
k^{2}
) of the data values are k standard deviations of the mean.
Important things to keep in mind about Chebyshev’s theorem
The distance between µ and µ + kσ is kσ, since µ + kσ – µ = kσ
The distance between µ and µ – kσ is kσ, since µ – (µ – kσ) = µ – µ – – kσ = 0 – – kσ = kσ
The answer to (1 –
1
k^{2}
) is usually expressed as a percentage after you finish calculating.
Notice that we are saying that k is greater than 1 or at least 2. k must be 2 because we get 0 when k = 1.
If k = 1 then (1 –
1 ) 
= (1 –
1 ) = 1 – 1 = 0 
How to find the standard deviation using Chebyshev’s theorem
Now let’s see how we can apply Chebyshev’s theorem.
For example, suppose you want to find the percentage of values in a data set that are within 2 standard deviations of the mean.
Just replace k = 2 in the formula.
= (1 –
1 ) = 1 – 0.25 = 0.75 
0.75 as percent is 75%
Therefore, 75% of the values in a data set are within 2 standard deviations of the mean.
How do we use the mean and standard deviation to find the percentage of total observations that fall within a given interval around the mean?
Assuming µ = 39 and σ = 5, find the percentage of the values that are between 29 and 49 of the mean.
29 ——– 39 ——————– 49
We just need to find k and there are 2 ways to do it.
First notice, as mentioned earlier, that the distance between 39 and 49 is equal to kσ
Since the distance between 39 and 49 is 10, kσ = 10
kσ = 10
σ = 5, so 5k = 10
Since 5 times 2 = 10, k = 2.
We calculated the percentage earlier when k = 2. We found it to be 75%.
Therefore, 75% of the values are between 29 and 49 of the mean.
The second To find the answer, you need to find that µ + kσ = 49
29 ————————– 39 ——————— — 49
µ – kσ µ µ + kσ
µ + kσ = 49
39 + kσ = 49
3939 + kσ = 4939
0 + kσ = 10
kσ = 10
5k = 10 and k = 2

Chebyshev’s theorem
May 31, 21:32 a.m.
What is Chebyshev’s theorem? Definition and simple examples to help you understand quickly.
Continue reading