Question

What does Chebyshev's theorem state?

Answer 1

Chebyshev's inequality is more general, stating that a minimum of just 75% of values must lie within two standard deviations of the mean and 88.89% within three standard deviations for a broad range of different probability distributions.

Chebyshev's Theorem estimates the minimum proportion of observations that fall within a specified number of standard deviations from the mean. This theorem applies to a broad range of probability distributions. Chebyshev's Theorem is also known as Chebyshev's Inequality.

If you have a mean and standard deviation, you might need to know the proportion of values that lie within, say, plus and minus two standard deviations of the mean. If your data follow the normal distribution, that's easy using the Empirical Rule! However, what if you don't know the distribution of your data or you know that it doesn't follow the normal distribution? In that case, Chebyshev's Theorem can help you out!

Maximum proportion of observations that are more than \(\mathrm{k}\) standard \[\frac{1}{k^{2}}\] deviations from the mean
Minimum proportion of observations that are within \(k\) standard deviations of the mean
\[
1-\frac{1}{k^{2}}
\]
Where \(k\) equals the number of standard deviations in which you are interested. \(K\) must be greater than \(1 .\)

Comments

At least \(\left(1-1 / z^{2}\right)\) of the data values must be within \(z\) standard deviations of the mean, where \(z\) is any value greater than \(1 .\)
At least \(75 \%\) of the data values must be within \(z=2\) standard deviations of the mean.

At least \(89 \%\) of the data values must be within \(z=3\) standard deviations of the mean.

At least \(94 \%\) of the data values must be within \(z=4\) standard deviations of the mean.