Chapter 8
Normal Distributions
■ sketch normal curves to illustrate distributions or
probabilities
■ use a normal distribution to model a continuous random
variable
Continuous Random Variables
• A continuous variable is a variable that can
assume any value on a continuum (can assume
an uncountable number of values)
 thickness of an item
 time required to complete a task
 temperature of a solution
 height, in inches
Continuous Random Variables
We talk about the probability of the random
variable assuming a value within a given
interval, NOT on a given particular value
When X is a discrete random variable, we can
represent 𝑷(𝑿 = 𝒓).
When X is a continuous random variable, we
can represent 𝑷(𝒂 ≤ 𝑿 ≤ 𝒃).
Properties of Normal Distributions
Normal distribution
• A continuous probability distribution for a random variable, x.
• The most important continuous probability distribution in
statistics.
• The graph of a normal distribution is called the normal curve.
x
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Properties of Normal Distributions
1. The mean, median, and mode are equal.
2. The normal curve is bell-shaped and is symmetric about the mean.
3. The total area under the normal curve is equal to 1.
4. The normal curve approaches, but never touches, the x-axis as it
extends farther and farther away from the mean.
Total area = 1
μ
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x
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Properties of Normal Distributions
5. Between μ – σ and μ + σ (in the center of the curve), the
graph curves downward. The graph curves upward to the
left of μ – σ and to the right of μ + σ. The points at which
the curve changes from curving upward to curving
downward are called the inflection points.
μ – 3σ
μ – 2σ
μ–σ
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μ
μ+σ
μ + 2σ
μ + 3σ
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Means and Standard Deviations
• A normal distribution can have any mean and any positive standard
deviation.
• The mean gives the location of the line of symmetry.
• The standard deviation describes the spread of the data.
μ = 3.5
σ = 1.5
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μ = 3.5
σ = 0.7
μ = 1.5
σ = 0.7
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 The mean gives the location of the line of symmetry.
 The standard deviation describes the spread of the data.
Example: Understanding Mean and Standard
Deviation
1. Which normal curve has the greater mean?
Solution:
Curve A has the greater mean (The line of symmetry of curve A
occurs at x = 15. The line of symmetry of curve B occurs at x = 12.)
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Example: Understanding Mean and Standard
Deviation
2. Which curve has the greater standard deviation?
Solution:
Curve B has the greater standard deviation (Curve B is more
spread out than curve A.)
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Example: Interpreting Graphs
The scaled test scores for the New York State Grade 8 Mathematics Test
are normally distributed. The normal curve shown below represents
this distribution. What is the mean test score? Estimate the standard
deviation.
Because the inflection points
Solution:
Because a normal curve is
symmetric about the mean, you
can estimate that μ ≈ 675.
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are one standard deviation
from the mean, you can
estimate that σ ≈ 35.
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Do Exercise 8A page 191-192
• Answer numbers 1a-f, 2a, 6
Homework (A4) Page 199
• Answer 8B numbers 1 a-f, 2 a-f
QUESTIONS?
Standard Normal Distribution
A normal distribution with a mean of 0 and
a standard deviation of 1.
Area = 1
–3
–2
–1
z
0
1
2
3
• Any x-value can be transformed into a z-score by
using the formula
Value  Mean
x
z

Standard deviation

Standard Normal Distribution
• If each data value of a normally distributed random variable x
is transformed into a z-score, the result will be the standard
normal distribution.
Normal Distribution
σ
z

x
x
Standard Normal
Distribution

σ1
0
z
• Use the Standard Normal Table to find the
cumulative area under the standard normal curve.
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Example (page 196)
Given that Z ~ N(0, 1), find P(Z < 1.23) and 𝑃(𝑍 ≥ 1.23)
Example (page 196)
Example (page 198)
Given that Z ~ N(0, 1), find the value of a such that
P(Z < a ) = 0.9072
Look at the table! closest to 0.9072
Example (page 196)
Given that Z ~ N(0, 1), find the value of b such that
𝑃 𝑍 ≥ 𝑏 = 0.7113
Homework (A4) Page 199
• Answer 8B numbers 1 a-f, 2 a-f
ANSWERS
The time spent by shoppers in a large shopping centre has a
normal distribution with mean 96 minutes and standard deviation 18
minutes.
(a) Find the probability that a shopper chosen at random spends
between 85 and 100 minutes in the shopping centre.
88% of shoppers spend more than t minutes in the shopping
centre.
(b) Find the value of t
Cambridge International AS & A Level Mathematics 9709 Paper 52 Q3 November 2021
Past Papers
Cambridge International AS & A Level Mathematics 9709 Paper 52 Q3 November 2021
Past Papers
Cambridge International AS & A Level Mathematics 9709 Paper 52 Q3 November 2021
Do Exercise 8A page 191-192
• Answer numbers
3 a, e, I
4 a, c, e
Standardising a Normal Distribution
Value  Mean
x
z

Standard deviation

Area = 1
–3
–2
–1
z
0
1
2
3
• Any x-value can be transformed into a z-score by
using the formula
Standardising a Normal Distribution
Standardising transforms the distribution X ~ N(µ, σ2 )
to Z ~ N(0, 1)
Useful Graphs
Example (page 201)
Given that X ~ N(11, 25), find P(X < 18) correct to 3
significant figures
Example (page 201)
Given that X ~ N(20, 7), find P(X ≤ 16.6) correct to 3
significant figures.
Example (page 201)
Given that X ~ N(5, 5), find 𝑃(2 ≤ 𝑧 < 9 correct to 3
significant figures
Example (page 202)
Given that Y ~ N(𝜇, 𝜎 2 ), P(Y <10) = 0.75 and
𝑃 𝑌 ≥ 12 = 0.1, find the values of µ and σ .
A mathematical puzzle is given to a large number of students.
The times taken to complete the puzzle are normally distributed with
mean 14.6 minutes and standard deviation 5.2 minutes.
(a) In a random sample of 250 of the students, how many would you
expect to have taken more than 20 minutes to complete the puzzle?
Cambridge International AS & A Level Mathematics 9709 Paper 51 Q4 May 2023
A mathematical puzzle is given to a large number of students.
The times taken to complete the puzzle are normally distributed with
mean 14.6 minutes and standard deviation 5.2 minutes.
All the students are given a second puzzle to complete. Their
times, in minutes, are normally distributed with mean - and standard
deviation 3. It is found that 20% of the students have times less than
14.5 minutes and 67% of the students have times greater than 18.5
minutes.
(b) Find the value of - and the value of 3.
Cambridge International AS & A Level Mathematics 9709 Paper 51 Q4 May 2023
Past Papers
• Cambridge International AS & A Level Mathematics 9709 Paper 51 Q4 May 2023
Past Papers
• Cambridge International AS & A Level Mathematics 9709 Paper 51 Q4 May 2023
Do Exercise 8C page 204
• Answer numbers
1 a, b
2 a, c, f
Homework (A4) Page 199
• Answer 8C numbers
3 a-d
4 a-d
7
9
11
12