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Normal distribution

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

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Normal distribution

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  1. Normal distribution And standard deviation

  2. 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

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

  4. 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%

  5. 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%

  6. 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

  7. 13.5% 13.5% 2% 0.5% 2% 0.5% 34% 34% 1 sd = 68% 2 sd = 95% 3 sd = 99% The probability of all possible events add to 100% or 1 • You can divide the normally distributed curve into sections

  8. 1 sd 2 sd 3 sd 915 925 935 955 965 975 945 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.

  9. 2 sd from the mean 2 sd 2 sd a. What percentage of pieces weigh between 925g and 965g? • How far from the mean are 925 and 965? This corresponds to 95% probability 915 925 935 955 965 975 945

  10. Data above the mean = 50% • Data from 935 to 945 = 34% (1 sd below mean) 1 sd 34% 50% 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 945 935

  11. 915 925 935 955 965 975 945 c. Below what weight will a piece of slate almost certainly be? • ‘Almost certainly’ corresponds to what sd? 3 sd Below 975g 1 sd 2 sd 3 sd

  12. 2 sd below mean 50% - 34% - 13.5% = 2.5% less than 925g 2 sd 50% 34% 13.5% 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? 2.5% x 400 10 pieces 915 925 935 945 955 965 975

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