029. Common Uses for the Standard Deviation

There are at least two additional applications for the standard deviation. The first one involves Chebyshev’s Theorem and applies to any distribution of observations, while the second is appropriate only if the distribution meets specific conditions of normality.

Chebyshev’s Theorem

The theorem was formulated by the Russian mathematician P. L.Chebyshev (1821 - 1894). It states that For Any data set, at least 1 – 1/K2 percent of the observations lie within K standard deviations of the mean, where K is any number greater than 1. Chebyshev’s Theorem is expressed as

. (3.19)

Thus, if statistician forms an interval from K = three standard deviations above the mean to three standard deviations below the mean, then at least of all observations will be within that interval.

Example 3.11. Passengers for P&P averaged 78.3 per day with (see Example 3.9) a standard deviation of 10.8. In order to schedule times for a new route P&P opened, management wants to know how often the number of passengers is within K = two standard deviations of the mean, and what that interval is.

Solution. Moving two standard deviations (2 x 10.8) = 21.6 passengers above and below the mean of 78.3, an interval of (78.3 – 21.6) = 56.7 to (78.3 + 21.6) = 99.9 passengers will be found. The management can be certain that at least of the time, the number of daily passengers was between 56 and 99.

Interpretation. On at least 75 percent of the days, the number of passengers was between 56 and 99. This provides the management of P&P with valuable information regarding how many passengers to prepare for in-flight operations.

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