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Normal distribution And standard deviation

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Normal Distributions Normally distributed data is described by giving the mean (the middle value) and the standard deviation (sd) or measure of spread The graph fits a bell-shaped curve with most measurements near the middle and a few extreme values above or below the mean

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Example The lifetime of a light bulb is normally distributed, with mean 900 hours, and standard deviation 150 hours. 900 mean sd sd sd

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Proportions 68% of the data lies within 1 standard deviation (either side) of the mean. It is likely or probable that the data will be in this region. 68% 1 sd

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Proportions 95% of the data lies within 2 standard deviations (either side) of the mean. It is very likely or very probable that the data will be in this region. 95% 2 sd

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Proportions 99% of the data lies within 3 standard deviations (either side) of the mean. It is almost certain that the data will be in this region. 99% 3 sd

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The probability of all possible events add to 100% or 1 You can divide the normally distributed curve into sections 34% 1 sd = 68% 2 sd = 95% 13.5% 0.5% 2% 3 sd = 99%

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Slate Flooring An architect chooses slate for a floor. Each piece varies in weight and are normally distributed with a mean of 945g and a standard deviation of 10g sd sd sd

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a. What percentage of pieces weigh between 925g and 965g? How far from the mean are 925 and 965? sd 2 sd from the mean This corresponds to 95% probability

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b. What percentage of pieces weigh more than 935g? Add everything above the mean to the section below the mean 84% weigh more than 935g Data above the mean = 50% 50% Data from 935 to 945 = 34% (1 sd below mean) 34% 1 sd

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c. Below what weight will a piece of slate almost certainly be? ‘Almost certainly’ corresponds to what sd? 3 sd Below 975g sd 2 sd 3 sd

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d. 400 pieces of slate are ordered for the floor. Estimate how many will weigh less than 925g Where is 925g situated on the curve? % x sd below mean 2 sd 50% - 34% % = 2.5% less than 925g 50% 34% 13.5% 10 pieces

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