How To: Estimate probabilities with the empirical rule

Estimate probabilities with the empirical rule

Learn how to use the empirical rule (or 68-95-99.7 rule) to estimate probabilities for normal distributions in statistics. From Ramanujan to calculus co-creator Gottfried Leibniz, many of the world's best and brightest mathematical minds have belonged to autodidacts. And, thanks to the Internet, it's easier than ever to follow in their footsteps (or just finish your homework or study for that next big test). With this installment from Internet pedagogical superstar Salman Khan's series of free math tutorials, you'll learn how to estimate probabilities for a bell distribution.

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what percentage of women have shoe sizes greater than 11.33 if the mean is 8.27 and the deviation is 1.53?

Since the larger shoe size is two standard deviations above the mean, one expects it is exceeded by about 2.5% of the shoe sizes sampled, if shoe sizes follow the normal distribution and the sample is large enough. (The 0.68, 0.95, 0.997 in the rule are the the areas within 1, 2, or 3 standard deviations of the mean. In each case, one-half the area located farther from the mean is in the high tail, on the right side of a bell curve graph. So 100% - 95% = 5%, of which half, or 2.5%, is in the high tail.) In general, however, the bell curve works best at predicting percentages that score in ranges close to the mean, and becomes less accurate for unusually high or low scores. Two standard deviations is as far as one should extrapolate a bell curve with most social science variables. It is also possible to reify the bell curve as IQ tests do: The way the tests are scored forces them to fit a bell curve, albeit with reproducible results. But from this we can't conclude that human intelligence, the thing IQ tests are supposed to measure, is normally distributed, or even linear for that matter. In fact there's good reason to think that intelligence isn't additive like linear functions are. The "smartest" people may enjoy only a slight edge in raw computational brain power over their "dumbest" peers; intellectual tasks being of a nature where a very few extra neural connections of the right kind have a big effect on results.

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