What is the standard deviation good for? Numerical data that come from an experiment carry inherently some “unit of measure”, for example How long a battery.

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Presentation transcript:

What is the standard deviation good for? Numerical data that come from an experiment carry inherently some “unit of measure”, for example How long a battery lasts is usually measured in months. How far from the target’s center an arrow hits is usually measured in inches. The weight of a human hair is usually measured in micrograms (10 -6 g.)

Since the standard deviation is computed by taking a square root of average distances squared, it is measured in the same units. The principal property of the standard deviation is that it allows us to estimate the proportion of data which are within certain distances from the mean. More precisely, if we take any interval centered at the mean, of width 2d (d is simply some positive number, half the length of the interval) then …

… the number (how many ‘s go into d) will allow to estimate the percentage of data falling inside the interval. There are two possibilities, either we know the data are nicely distributed (the histogram is mound-shaped and approximately symmetric), or we do not. If we do know, then the rule known as The Empirical Rule applies. Otherwise (i.e., always, but weaker) we apply Tchebyshev’s Theorem

The tables below show both: