(rev. 9/5/2000, 2/3/2007)

Chebyshev’s Theorem (CT) is more general than the Normal Rule. Basically, Chebyshev’s Theorem states that in any distribution with finite standard deviation, as long as the value x that you have in mind satisfies x > 1, the fraction of observations that fall within x s.d.'s of the mean is at least 1 – (1/x²).

For example, regardless of the shape of a distribution, at least 84% of the data points must always fall within 2.5 s.d.’s of the mean. Why is this so? Apply CT with x = 2.5. Since x > 1, at least 1 – (1/2.5²), or 0.84, or 84%, of the observations must fall within 2.5 standard deviations of the mean.

Does CT contradict the Normal Rule? No, because CT says that at least a certain fraction of values must fall within so-and-so-many s.d.’s of the mean. With normal distributions, the requirement is met with plenty of room to spare.

For x = 2, CT guarantees at least 75% of values within 2 s.d.’s of the mean. The Normal Rule says 95%, which is certainly greater than 75%.

For x = 3, CT guarantees at least 88.9% of the values within 3 s.d.’s of the mean. The Normal Rule says 99.7%, which is certainly greater than 88.9%.