1. Normal Distribution

2. Standard Deviation Calculate the mean Given a Data Set 12, 8, 7, 14, 4
= 9
The standard deviation is a measure of the mean spread of the data from the mean.
How far is each data value from the mean?
( ) ÷ 5 = 12.8
Square to remove the negatives
Square root 12.8 = 3.58 25 1 9 4
Std Dev = 3.58 4 7 14 12 8 -2 -1 3 5 -5
Average =
Sum divided by how many values
Calculator function
Square root to ‘undo’ the squared

3. The Normal Distribution
Key Concepts
Area under the graph is the relative frequency = the probability
Total Area = 1
The MEAN is in the middle.
The distribution is symmetrical.
1 Std Dev either side of mean = 68%
A lower mean
A higher mean
2 Std Dev either side of mean = 95%
3 Std Dev either side of mean = 99%
A smaller Std Dev.
A larger Std Dev.
Distributions with different spreads have different STANDARD DEVIATIONS

4. Find the probability a chicken is less than 4kg
Finding a Probability
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
4kg
3kg
Draw a distribution graph 1
How many Std Dev from the mean?
4kg
distance from mean
standard deviation
= = 2.5 1
0.4
3kg
Look up 2.5 Std Dev in tables (z = 2.5)
0.5
0.4938
Probability = (table value) =
4kg
3kg
So 99.38% of chickens in the population weigh less than 4kg

5. Standard Normal Distribution
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
3kg
2.6kg
Draw a distribution graph
Table value
0.5
Change the distribution to a Standard Normal
distance from mean
standard deviation
z =
= = 1.333
0.4
0.3 z
= 1.333
Aim: Correct Working
The Question:
P(x < 3kg)
= P(z < 1.333)
Look up z = Std Dev in tables =
Z = ‘the number of standard deviations from the mean’ =

6. Inverse Normal Distribution
The mean weight of a chicken is 2.6 kg (with a standard deviation of 0.3 kg)
2.6kg
‘x’ kg
Area = 0.9
90% of chickens weigh less than what weight? (Find ‘x’)
Draw a distribution graph
0.4
0.5
Look up the probability in the middle of the tables to find the closest ‘z’ value.
Z = ‘the number of standard deviations from the mean’ z
= 1.281
The closest probability is
Look up 0.400
Corresponding ‘z’ value is: 1.281
z = 1.281
2.6kg D
D = × 0.3
The distance from the mean = ‘Z’ × Std Dev
2.98 kg
x = 2.6kg = kg