The empirical rule for the SD

Another interpretation of the SD involves the empirical rule for the SD , which we state as follows.

When we say that Pharmacia stock prices averaged $50.54 the past two years give or take $6.82, we may expect approximately 68% of the observations (14 out of 21) to fall within $50.54 $\pm$ $6.82. In fact, 13 out of 21 observations do. We expect approximately 95% of the observations (20 out of 21) to fall within $50.54 $\pm$ 2($6.82). In fact, all 21 observations do.

The empirical rule is a good way to come up with an ``educated guess'' of the average and SD of a process. For example, suppose you were asked about the price of gas in Kalamazoo. It is easy to come up with a ``typical" gas price, say, around $1.69 during the academic year 2002-2003. It is harder to come up with a measure of variability: $1.69 plus or minus ________ ? However, if you can come up with some estimate of an interval within which gas prices fell most of the time, you can deduce a standard deviation from this. Suppose that approximately 19 times out of 20 (or 95% of the time), gas prices during the year fell between $1.39 and $1.79, this implies an average of $1.59 with an SD of 10 cents.