1. And standard deviation
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. 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

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. 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%

8. 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

9. 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

10. 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

11. 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

12. 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% % = 2.5% less than 925g
50%
34%
13.5%
2.5% x 400
10 pieces
915
925
935
945
955
965
975