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The Empirical Rule, Chebyshev’s Theorem, Percentiles and z-scores Today, we will study The Empirical Rule and Chebyshev’s Theorem and how to apply them. The Empirical Rule The Empirical Rule: •68% of the data under a normal curve lies within one standard deviation of the mean. •95% of the data under a normal curve lies within two standard deviations of the mean. •99.7% of the data under a normal curve lies within one standard deviation of the mean. 68% 95% 99.7% The Empirical Rule Here is a another way to look at it: 34% 34% 2.35 2.35 13.5% 13.5% The Empirical Rule The Empirical Rule: A normal population has a mean of 50 and a standard deviation of 5, between what two numbers does 95% of the data lie. Answer: •Between 2 standard deviations from the mean. 50 + 2*5 = 60 50 – 2*5 = 40 The Empirical Rule The Empirical Rule: A normal population has a mean of 50 and a standard deviation of 5, what percent of the data lies between 45 and 50 Answer: •68% of the data lies within 1 sd of the mean. •The percent between 45 and 50 is half of that: 68/2 = 34% The Empirical Rule The Empirical Rule: A normal population has a mean of 50 and a standard deviation of 5, what percent of the data lies between 40 and 45 Answer: •½ of 95% = 47.5% of the data lies between 40 and 50. •½ of 68% = 34% of the data lies between 45 and 50. •47.5 – 34 = 13.5% of the data between 40 and 45 The Empirical Rule Remember the Empirical Rule only applies to the normal distribution. 68% 95% 99.7% Chebyshev’s Rule For K> 1, the portion of any data set lying within K standard deviations of the mean is at least: 1 1 2 k For k = 2: 1 3 1 2 2 4 For k = 3: 1 8 1 2 3 9 Chebyshev’s Rule A non-normal data set has a mean of 50 and a standard deviation of 5, between what two numbers does 75% of the data lie? We can’t use the Empirical Rule for nonnormal data, so we use Chebyshev’s: 75% of the data is within two standard deviations of the mean. So, 75% of the data is between 40 and 60.