The Normal Distribution Worksheet - Oxford Insight Mathematics General Hsc General 2 Student Book Page 10
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206D
Properties of the normal distribution
Properties of the normal distribution
The normal distribution has been studied in some detail by mathematicians and it can be shown that for a normal
distribution, approximately:
• 68% of the scores will lie within one standard deviation of the mean: −1 < z-score < 1
• 95% of the scores will lie within two standard deviations of the mean: −2 < z-score < 2
• 99.7% of the scores will lie within three standard deviations of the mean: −3 < z-score < 3
This may be illustrated as follows.
68%
95%
99.7%
x − s
x + s
x − 2s
x + 2s
x − 3s
x + 3s
x
x
x
WORKED EXAMPLE 1
The marks on a test are normally distributed with a mean of 65 and a standard deviation of 8.
a
What mark is one standard deviation:
i
ii
above the mean?
below the mean?
b
What proportion of students scored a mark between 57 and 73?
c
What mark is two standard deviations:
i
ii
above the mean?
below the mean?
d
What proportion of students scored a mark between 49 and 81?
e
What mark is three standard deviations:
i
ii
above the mean?
below the mean?
f
What proportion of students scored a mark between 41 and 89?
Solve
Think
Apply
65 + 8 = 73
a
i
Add and subtract the standard
One standard deviation
deviation.
above the mean is a
65 − 8 = 57
ii
z-score of +1.
b
Approximately 68% of the students
68% of scores are between
One standard deviation
z-scores of ±1.
scored a mark between 57 and 73.
below the mean is
65 + 2 × 8 = 81
a z-score of −1.
c
i
Add and subtract 2 times standard
deviation from the mean.
Similarly for z-score
65 − 2 × 8 = 49
ii
of ±2, ±3.
These are z-scores of ±2,
d
Approximately 95% of the students
∴ 95%.
scored a mark between 49 and 81.
65 + 3 × 8 = 89
e
i
Add and subtract 3 times the
standard deviation from the mean.
65 − 3 × 8 = 41
ii
These are z-scores of ±3,
f
Approximately 99.7% of the students
∴ 99.7%.
scored a mark between 41 and 89.
208
Insight Mathematics General 12
HSC Course 2
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