Normal distributionAnd 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

1050750

1 sd1 sd

450 1350

3 sd 3 sd

1200600

2 sd 2 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.

68%

1 sd

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

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

34% 34%

1 sd = 68%

2 sd = 95%

13.5% 13.5%

0.5%0.5% 2%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.

945 955935

1 sd

965925

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?

915 975965955935925 945

2 sd2 sd

2 sd from the mean

This corresponds to 95% probability

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%

945935

• Data from 935 to 945 = 34% (1 sd below mean)

34%

1 sd

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

3 sd

Below 975g

915 975965955935925 945

1 sd

2 sd

3 sd

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?

925 935 945 965955

2.5% x 400

915 975

2 sd below mean

2 sd

50% - 34% - 13.5% = 2.5% less than 925g

50%

34%13.5%

10 pieces