Chebychev and Empirical Rules

Knowing the mean and standard deviation of a sample or a population gives us a good idea of where most of the data values are because of the following two rules:

• 's Rule The proportion of observations within k standard deviations of the mean, where , is at least , i.e., at least 75%, 89%, and 94% of the data are within 2, 3, and 4 standard deviations of the mean, respectively.
• Empirical Rule If data follow a bell-shaped curve, then approximately 68%, 95%, and 99.7% of the data are within 1, 2, and 3 standard deviations of the mean, respectively.

EXAMPLE: A pharmaceutical company manufactures vitamin pills which contain an average of 507 grams of vitamin C with a standard deviation of 3 grams. Using Chebychev's rule, we know that at least

or 75% of the vitamin pills are within k=2 standard deviations of the mean. That is, at least 75% of the vitamin pills will have between 501 and 513 grams of vitamin C, i.e.,

EXAMPLE: If the distribution of vitamin C amounts in the previous example is bell shaped, then we can get even more precise results by using the empirical rule. Under these conditions, approximately 68% of the vitamin pills have a vitamin C content in the interval [507-3,507+3]=[504,510], 95% are in the interval [507-2(3),507+2(3)]=[501,513], and 99.7% are in the interval [507-3(3),507+3(3)]=[498,516].

NOTE: Chebychev's rule gives only a minimum proportion of observations which lie within k standard deviations of the mean.