12 FURTHER MATHEMATICS
The Mean and the Standard Deviation
The Mean and the Standard Deviation
The Mean and the Standard Deviation
The Mean and the Standard Deviation
The Mean and the Standard Deviation
The Mean and the Standard Deviation
The Mean and the Standard Deviation
The Mean and the Standard Deviation
The Mean and the Standard Deviation
The Mean and the Standard Deviation
The Mean and the Standard Deviation
The Mean and the Standard Deviation
The Mean and the Standard Deviation
The Normal Distribution
The Normal Distribution
Standard deviation is a measure of the spread of a data
distribution about the average (mean)
For normal distributions, we can determine the % of
values that lie within 1, 2 or 3 standard deviations of
the mean:
The Normal Distribution

68% of values lie within 1 standard deviation
of the mean (with 16% of values either side)
The Normal Distribution

95% of values lie within 2 standard deviations of the mean
(with 2.5% of values either side)
The Normal Distribution

99.7% of values lie within 3 standard deviations of the
mean (with 0.15% of values either side)
The Normal Distribution

50% of values lie either side of the mean
Summary Diagram

The diagram below can be used in all cases. It shows the % of
values that lie in each section of the normal distribution curve
Example 1
We always start by drawing diagrams. Each diagram below has been
scaled with a mean of 134 and standard deviation of 20
Diagram 1
Diagram 3
114 134 154
Diagram 2
94
74
134
Diagram 4
134
174
134
194
The Normal Distribution
Example 1 – solutions
a) About 68% of executives have blood pressure between 114 and 154 (diagram 1)
b) About 95% of executives have blood pressure between 94 and 174 (diagram 2)
c) About 99.7% of executives have blood pressure between 74 and 194 (diagram 3)
d) About 16% of executives have blood pressures above 154 (diagram 1)
e) About 2.5% of executives have blood pressures below 94 (diagram 2)
f) About 0.15% of executives have blood pressures below 74 (diagram 3)
g) About 50% of executives have blood pressures above 134 (diagram 4)
The Normal Distribution

Example 2
For this question we could again use the four separate diagrams that were used in example
1, but instead we can use the summary diagram (see next page). Again a scale is included
for a mean of 170 and standard deviation of 5
The Normal Distribution
The Normal Distribution
Example 2 – solutions
a)
vi.
Between 155cm and 185cm: 2.35% + 13.5% + 34% + 34% + 13.5% + 2.35% = 99.7%
More than 175cm: 13.5% + 2.35% + 0.15% = 16%
More than 170cm: 34% + 13.5% + 2.35% + 0.15% = 50%
Less than 160cm: 2.35% + 0.15% = 2.5%
Less than 165cm: 13.5% + 2.35% + 0.15% = 16%
Between 160cm and 180cm: 13.5% + 34% + 34% + 13.5% = 95%
b)
We already found that 16% are likely to have heights above 170cm. 16% of 5000 = 0.16 x 5000 = 800
i.
ii.
iii.
iv.
v.
So 800 women are expected to have heights above 170cm
The Normal Distribution

The advantage of the summary chart is for a question like this (not
symmetrical):
What % of these women have heights between 165cm and 180cm?
Answer: 34% + 34% + 13.5% = 81.5%
81.5% of women have heights between 165cm and 180cm
Z-scores
We can also determine how many standard
deviations above or below the mean a value lies by
standardising the data. The standardised scores
are called z-scores.
To calculate z-score:
Example 1
A set of data has a mean of 65 and a
standard deviation of 6. Standardise
the following scores:
a)
b)
c)
53
68
75
WORK TO BE COMPLETED
Chapter 3