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Normal Distribution
Links
Standard Deviation
The Normal Distribution
Finding a Probability
Standard Normal Distribution
Inverse Normal Distribution
Standard Deviation
Calculate the mean
Mean = (12 + 8 + 7 + 14 + 4) ÷ 5
x =9
Given a Data Set
12, 8, 7, 14, 4
The standard deviation is a
measure of the mean spread of
the data from the mean.
(25 + 4 + 25 + 1 + 9) ÷ 5 = 12.8
Square root 12.8 = 3.58
25
-5
7
4
n
Calculator 8
function
4
5
6
7
Std Dev = 3.58
4
-2
25
5
1
-1
8
9
3
12
How far is each data
value from the mean?
xx
x  x 
2
 x  x 
2
n
14
9 10 11 12 13 14
x
1st slide
Square to remove
the negatives
Average =
Sum divided by
how many values
 x  x 
2

n
Square root
to ‘undo’ the
squared
The Normal Distribution
Key Concepts
1st slide
Area under the graph is the
relative frequency = the probability
Total Area = 1
The MEAN is in the middle.
The distribution is symmetrical.
x
A lower
mean
1 Std Dev either
side of mean = 68%
A higher
mean
x
x
A smaller
Std Dev.
2 Std Dev either
side of mean = 95%
3 Std Dev either
side of mean = 99%
A larger
Std Dev.
Distributions with different
spreads have different
STANDARD DEVIATIONS
Finding a Probability
1st slide
The mean weight of a chicken is 3 kg
(with a standard deviation of 0.4 kg)
Find the probability a
chicken is less than 4kg
x
4kg
3kg
Draw a distribution graph
1
How many Std Dev from the mean?
x
1
distance from mean
=
= 2.5
0.4
standard deviation
Look up 2.5 Std Dev in tables (z = 2.5)
Probability = 0.5 + 0.4938 (table value)
= 0.9938
4kg
3kg
0.5
0.4938
x
3kg
So 99.38% of chickens in the population weigh less than 4kg
4kg
Standard Normal Distribution
1st slide
The mean weight of a chicken is 2.6 kg
(with a standard deviation of 0.3 kg)
Find the probability a
chicken is less than 3kg
x
3kg
2.6kg
Draw a distribution graph
Table
value
0.5
Change the distribution to a Standard Normal
z=
distance from mean
0.4
=
= 1.333
standard deviation
0.3
x x
z 

0
z = 1.333
Aim: Correct Working
The Question: P(x < 3kg)
= P(z < 1.333)
Look up z = 1.333 Std Dev in tables
Z = ‘the number of standard
deviations from the mean’
= 0.5 + 0.4087
= 0.9087
Inverse Normal Distribution
The mean weight of a chicken is 2.6 kg
(with a standard deviation of 0.3 kg)
1st slide
Area = 0.9
90% of chickens weigh less
than what weight? (Find ‘x’)
x
‘x’ kg
2.6kg
Draw a distribution graph
Look up the probability in the middle of
the tables to find the closest ‘z’ value.
0.5
0.4
Z = ‘the number of standard
deviations from the mean’
0
The closest probability is 0.3999
Look up 0.400
Corresponding ‘z’ value is: 1.281
z = 1.281
The distance from the
mean = ‘Z’ × Std Dev
D = 1.281 × 0.3
z = 1.281
D
x
2.98 kg
x = 2.6kg + 0.3843 = 2.9843kg 2.6kg