What are the differences between the Emperical Rule and Chebyshev's Theorem?

Answer 1

The Empirical Rule applies solely to the NORMAL distribution, while Chebyshev's Theorem (Chebyshev's Inequality, Tchebysheff's Inequality, Bienaymé-Chebyshev Inequality) deals with ALL (well, rather, REAL-WORLD) distributions. The Empirical Rule is stronger than Chebyshev's Inequality, but applies to fewer cases.

The Empirical Rule:

- Applies to normal distributions.

- About 68% of the values lie within one standard deviation of the mean.

- About 95% of the values lie within two standard deviations of the mean.

- About 99.7% of the values lie within three standard deviations of the mean.

- For more precise values or values for another interval, use a normalcdf function on a calculator or integrate e^(-(x - mu)^2/(2*(sigma^2))) / (sigma*sqrt(2*pi)) along the desired interval (where mu is the population mean and sigma is the population standard deviation).

Chebyshev's Theorem/Inequality:

- Applies to all (real-world) distributions.

- No more than 1/(k^2) of the values are more than k standard deviations away from the mean. This yields the following in comparison to the Empirical Rule:

- No more than [all] of the values are more than 1 standard deviation away from the mean.

- No more than 1/4 of the values are more than 2 standard deviations away from the mean.

- No more than 1/9 of the values are more than 3 standard deviations away from the mean.

- This is weaker than the Empirical Rule for the case of the normal distribution, but can be applied to all (real-world) distributions. For example, for a normal distribution, Chebyshev's Inequality states that at most 1/4 of the values are beyond 2 standard deviations from the mean, which means that at least 75% are within 2 standard deviations of the mean. The Empirical Rule makes the much stronger statement that about 95% of the values are within 2 standard deviations of the mean. However, for a distribution that has significant skew or other attributes that do not match the normal distribution, one can use Chebyshev's Inequality, but not the Empirical Rule.

- Chebyshev's Inequality is a "fall-back" for distributions that cannot be modeled by approximations with more specific rules and provisions, such as the Empirical Rule.