Year 12 General Mathematics - HSC Course - Lesson Plans
Topic – The Normal Distribution (DA6)
Chapter 6
Lesson 1
Standard Scores - Intro
What are Standard Scores?
 Standard scores are a way of measuring the quality of a score relative to the mean and standard deviation
of each test.
 Read all of p236 together…
Raw score  mean
or
standard deviation
xx
On the formula sheet
z
, where s = n for a population standard deviation.
s
 Standard scores can compare scores from different tests and rate the quality of each score.
Standardised Score
DA6  The Normal Distribution
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Example 1
In a Maths exam the mean was 60 and the standard deviation was 10.
(a)
Calculate the z-score (standardised scores) for marks of:
(i)
50
(ii)
58
(iii) 60
(iv)
64
(v)
83
Answers:
(b)
(i)
z
(ii)
z
(iii)
z
(iv)
z
(v)
z
xx
s
xx
s
xx
s
xx
s
xx
s





50  60
10
58  60
10
60  60
10
64  60
10
83  60
10
= -1
= -0.2
=0
= 0.4
= 2.3
Explain the meaning of these scores.
Answers:
(i)
(ii)
(iii)
(iv)
(v)
50 is one (1) standard deviation below the mean,  z = -1
2
58 is 2 marks below the mean which is 10 (0.2) standard deviations below the mean,
 z = -0.2
60 is the exact mean,  z = 0.
4
64 is 4 marks above the mean which is 0.4 (10 ) standard deviations above the mean,
 z = 0.4
83 is 23 marks above the mean which is 2.3 standard deviations above the mean,
 z = 2.3
Example 2
Write down the z-score corresponding to a raw score which is:
(a)
08 standard deviations above the mean
(b)
equal to the mean
(c)
1.3 standard deviations below the mean
Answers:
(a)
(b)
(c)
z = 0.8
z=0
z = -1.3
DA6  The Normal Distribution
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Example 3
James scored 80 in an English test in which the mean was 74 and the standard deviation 8. He scored 52 in a
Maths test in which the mean was 48 and the standard deviation was 9.
(a)
Calculate his standardised score for each test.
(b)
In which class did he perform better?
(c)
What mark would James need to score in English for his performance to be equivalent to that in
Maths?
x  x 80  74

= 0.75
s
8
x  x 52  48

For Maths z 
= 0.44
s
9
(a)
For English z 
(b)
He performed better in English!
(c)
To be equivalent in English, a z-score of 0.44 is needed.
Mark = Mean + 0.44  Standard deviation
= 74 + 0.44  8
= 77.6
Class Work
Exercise 6A  Q1-10
DA6  The Normal Distribution
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Student Worksheet – Standard Scores
Intro - What are Standard Scores?
Raw score  mean
or
standard deviation
xx
On the formula sheet
z
, where s = n for a population standard deviation.
s
 Standard scores can compare scores from
tests and rate the
of each score.
Example 1
In a Maths exam the mean was 60 and the standard deviation was 10.
(a)
Calculate the standard scores for marks of:
(i)
50
(ii)
58
(iii) 60
(iv)
64
(v)
83
Answers:
(i)

Standardised Score
=
(ii)
(iii)
(iv)
(b)
(v)
Explain the meaning of these scores.
(i)
(ii)
(iii)
(iv)
(v)
Example 2
Write down the z-score corresponding to a raw score which is:
(a)
08 standard deviations above the mean
(b)
equal to the mean
(c)
1.3 standard deviations below the mean
Example 3
James scored 80 in an English test in which the mean was 74 and the standard deviation 8. He scored 52 in a
Maths test in which the mean was 48 and the standard deviation was 9.
(a)
Calculate his standardised score for each test.
(b)
In which class did he perform better?
(c)
What mark would James need to score in English for his performance to be equivalent to that in
Maths?
(a)
For English
For Maths
(b)
He performed better in
(c)
To be equivalent in English, a z-score of ________is needed.
Mark = Mean + _________  Standard deviation
=
=
Class Work - Exercise 6A  Q1-10
DA6  The Normal Distribution
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Lesson 2 – Frequency Graphs
Frequency Graphs
 Frequency graphs include Histograms and Polygons. Sometimes a smooth curve can be drawn to which
we can make judgements about the data.
 Read Example 1, p205 together!
Look at some questions from Exercise 6B and their Answers together…
DA6  The Normal Distribution
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Lesson 3 – The Normal Distribution
 Read through p206 (below)

For a Normal Distribution,
 The frequency curve is bell shaped and symmetrical about the mean.
 The mean, median and mode are equal.
Class Work
Exercise 6C Q1-3
DA6  The Normal Distribution
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Properties of the Normal Distribution – Student Worksheet
 For a normal distribution, approximately,
 68% of the scores will lie within 1 standard deviation of the mean,
i.e. ___ < z-score < ___
 95% of the scores will lie within 2 standard deviations of the mean,
i.e. ___ < z-score < ___
 99.7% of the scores will lie within 3 standard deviations of the mean,
i.e. ___ < z-score < ___
 This can be illustrated as follows… (this information is ON THE FORMULA SHEET)
See the link:
http://www.mathsisfun.com/data/standard-normal-distribution.html
Example 1
The marks on are test are normally distributed with a mean of 65 and a standard deviation of 10.
(a)
What marks are one s.d. above and below the mean? What proportion of students scored between
these two marks?
(b)
What marks are two s.d. above and below the mean? What proportion of students scored between
these two marks?
(c)
What marks are three s.d. above and below the mean? What proportion of students scored between
these two marks?
Example 2
Darren scored 57 in a test where the marks were normally distributed. The mean was 65 and the standard
deviation was 8. What percentage of students scored a better mark than Darren?
___
___%
___
___% ___% 
___
___%
___
 ________%   _______% 
___
____________%
_____%
___
  _____________% 
Darren’s result was ____
There are ___% below the mean that have a better result.
There are ___% above the mean that have a better result.
There are ____% who have a better result!
Class Work
Exercise 6D  Q1,2,3,4,5,6,8,9,11(a,b)
 Diagnostic Test + Review Set 6A complete…
DA6  The Normal Distribution
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Lesson 4 – Properties of the Normal Distribution (TEACHER VERSION)
 For a normal distribution, approximately,
 68% of the scores will lie within 1 standard deviation of the mean,
 95% of the scores will lie within 2 standard deviations of the mean,
 99.7% of the scores will lie within 3 standard deviations of the mean,
 This can be illustrated as follows:
i.e. 1 < z-score < 1
i.e. 2 < z-score < 2
i.e. 3 < z-score < 3
See the link:
http://www.mathsisfun.com/data/standard-normal-distribution.html
Example 1
The marks on are test are normally distributed with a mean of 65 and a standard deviation of 10.
(a)
What marks are one s.d. above and below the mean? What proportion of students scored between
these two marks?
65±10 = 55-75 marks (68% of students)
(b)
What marks are two s.d. above and below the mean? What proportion of students scored between
these two marks?
65±20 = 45-85 marks (95% of students)
(c)
What marks are three s.d. above and below the mean? What proportion of students scored between
these two marks?
65±30 = 35-95 marks (99.7% of students)
Example 2
Darren scored 57 in a test where the marks were normally distributed. The mean was 65 and the standard
deviation was 8. What percentage of students scored a better mark than Darren?
57
68%
73
34%34% 
49
 47.5%
41

49.85%
95%
81
  47.5% 
99.7%

49.85%
89

Darren’s result
There are 34% below the mean that have a better result.
There are 50% above the mean that have a better result.
There are 84% who have a better result!
Class Work
Exercise 6D  Q1,2,3,4,5,6,8,9,11(a,b)
 Diagnostic Test + Review Set 6A complete…
DA6  The Normal Distribution
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DA6  The Normal Distribution
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Revision from Chapter 4 - Shape of Frequency Graphs
 To describe a graph you can mention smoothness, symmetry, number of modes.
For example,


Graph A
Graph B
Graph C
Graph D
Graph A is a smooth curve while Graph B is not smooth.
Graph C is unimodal while Graph D is bimodal (two modes). If a curve has more than two modes it is
called multimodal.
Graph E

Graph F
Graph E is negatively skewed (this may mean the test was too easy where most students have scored
high marks. Graph F is positively skewed. See Chapter 4 for further curves.
http://www.mathsisfun.com/data/skewness.html
DA6  The Normal Distribution
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
Read through Example 1,2, p244,245.
Example 1
For the data in the stem-and-leaf plot,
Stem
0
1
2
3
4
5
(a)
(b)
(c)
Leaf
5
8
07899
1233
8
determine whether the scores are positively/negatively skewed or symmetrical.
Are there any outliers? What are they?
Are there any clusters? What are they?
DA6  The Normal Distribution
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