We have seen that relative frequencies converge on theoretical probabilities. But how fast? When can we begin to use an observed relative frequency as a reliable estimate of a probability? This chapter gives some answers. They are a little more technical than most of this book. For practical purposes, all you need to know is how to use is the three boxed Normal Facts below.
EXPERIMENTAL BELL-SHAPED CURVES
On page 191 we had the result of a coin-tossing experiment. The graph was roughly in the shape of a bell. Many observed distributions have this property.
Example: Incomes. In modern industrialized countries we have come to expect income distributions to look something like Curve 1 on the next page, with a few incredibly rich people at the right end of the graph. But in feudal times there was no middle class, so we would expect the income distribution in Curve 2. It is “bimodal”–it has two peaks.
Example: Errors. We can never measure with complete accuracy. Any sequence of “exact” measurements of the same quantity will show some variation. We often average the results. We can think of this as a sample mean. A good measuring device will produce results that cluster about the mean, with a small sample standard deviation. A bad measuring device gives results that vary wildly from one another, giving a large standard deviation.
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