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.
