Emperical Rule

Emperical Rule - The Empirical Rule Sometimes the...

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1 The Empirical Rule Sometimes the distribution is symmetric and bell shaped. For these kind of distribution mean and sd together can describe the distribution fairly well. Most of the observations lie near the center or mean of the data. EMPIRICAL RULE summarizes the distribution

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2 The Empirical Rule For data with a bell-shaped distribution Approximately 68% of the observations will be within ONE STANDARD DEVIATION of the mean. mean 68%
3 Two Standard Deviations For data having a bell-shaped distribution Approximately 95% of the data values will be within TWO STANDARD DEVIATION of the mean. 2 SD 2 SD mean 95%

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4 Three Standard Deviations For data having a bell-shaped distribution Approximately 99.7% of the data values will be within THREE STANDARD DEVIATION of the mean. mean 3 SD 3 SD
5 Chebyshev’s Rule Applies to any distribution More general but less “precise” It says that for any number k > 1 , at least (1-1/k 2 ) fraction of the data will lie within k standard deviations of mean.

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6 Different k’s 1-1/3 2 = 0.89 1-1/2 2 = 0.75 Distance from the Mean Minimum Proportion of Values Falling Within Distance Number of Standard Deviations k = 2 k = 3 k = 4 1-1/4 2 = 0.94
Example 8.0 (Empirical Rule) In a large class of 500 students mean score is 84 and the standard deviation is 5. Suppose the distribution is bell shaped. Then by empirical rule: Approximately 68% of the students got between 79 and 89 Approximately 95% of the students will have score between 74 and 94 Approximately 99.7% of the students will get between 69 and 99

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Emperical Rule - The Empirical Rule Sometimes the...

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