And standard deviation

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Presentation transcript:

And standard deviation Normal distribution And standard deviation

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

Example The lifetime of a light bulb is normally distributed, with mean 900 hours, and standard deviation 150 hours. 900 mean 1050 750 1 sd 1200 600 2 sd 450 1350 3 sd

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. 1 sd 68%

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. 2 sd 95%

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

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

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. 955 935 1 sd 945 965 925 2 sd 915 975 3 sd

a. What percentage of pieces weigh between 925g and 965g? How far from the mean are 925 and 965? 2 sd 2 sd from the mean This corresponds to 95% probability 915 925 935 945 955 965 975

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

c. Below what weight will a piece of slate almost certainly be? ‘Almost certainly’ corresponds to what sd? 3 sd Below 975g 915 975 965 955 935 925 945 1 sd 2 sd 3 sd

d. 400 pieces of slate are ordered for the floor 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? 2 sd below mean 2 sd 50% - 34% - 13.5% = 2.5% less than 925g 50% 34% 13.5% 2.5% x 400 10 pieces 915 925 935 945 955 965 975