Tchebychev's inequality says that

p(|x- µ | > k(sigma) ) < 1/(k²)

where (sigma) is the standard deviation, and "p" denotes probability, or
relative frequency.

In particular, it says that

- at most 1/4 of the data is more than two standard deviation units from the mean (or at least 3/4 of the data is within two standard deviation units of the mean)
- at most 1/9 of the data is more than three standard deviation units from the mean (or at least 8/9 of the data is within three standard deviation units of the mean)
- at most 1/16 of the data is more than four standard deviation units from the mean (or at least 15/16 of the data is within four standard deviation units of the mean)

p(|x- µ | < k(sigma) ) > (1 - 1/(k²))

The result is not restricted to integer values of k. The proof follows from the observation that, e.g., if 1/4 of the data is exactly 2 (sigma) from the mean, that data alone will account for the entire variance ( (1/4)(2(sigma))² = (sigma)² )

Note that if at most 1/4 of the data is 2 or more standard deviation units from the mean, then at most 1/4 of the data is 2 or more standard deviation units above the mean.

Note also that Tchebychev's inequality provides a very loose bound, in general data will be much closer to the mean than required by Tchebychev's inequality.

Example: If the mean height is 68 inches, and the standard deviation is 4 inches, what fraction of the population can be taller than 74 inches?

(74-68)/4 = 1.5; 1/(1.5)² = .44. Hence at most 44% of the population can be taller than 74 inches by Tchebychev's inequality. [If height were normally distributed, less than 6% of the people would be taller than 74 inches, which is even much less than 22% which one would get from invoking symmetry (44% 1.5(sigma) away "should" have 22% 1.5(sigma) above and 22% 1.5(sigma) below).

**Competencies:** If the mean weight of students is 145 pounds with a standard deviation of 25 pounds, at most what fraction of the class can weigh more than 200 pounds? If there are 35 students in the class, at most how many can weigh more than 200 pounds?

**Reflection:**

**Challenge:** If the average weight of students is 145 pounds, what can you say about the fration of the class that weighs over 25 pounds (the standard deviation is not known)?