# Standard Normal Distribution(Edexcel International A Level Maths: Statistics 1)

Author

Dan

Expertise

Maths

## Standard Normal Distribution

#### What is the standard normal distribution?

•  The standard normal distribution is a normal distribution where the mean is 0 and the standard deviation is 1
• It is denoted by Z

#### Why is the standard normal distribution important?

• Calculating probabilities for the normal distribution can be difficult and lengthy due to its complicated probability density function
• The probabilities for the standard normal distribution have been calculated and laid out in the table of the normal distribution which can be found in your formula booklet
• Nowadays, many calculators can calculate probabilities for any normal distribution, if yours does then the tables can be used as a check
• It is possible to map any normal distribution onto the standard normal distribution curve
• Mapping different normal distributions to the standard normal distribution allows distributions with different means and standard deviations to be compared with each other

#### How is any normal distribution mapped to the standard normal distribution?

• Any normal distribution curve can be transformed to the standard normal distribution curve by a horizontal translation and a horizontal stretch
• Therefore, for  and , we have the relationship:

• Probabilities are related by:
•
• This is a very useful relationship for calculating probabilities for any normal distribution
• As the normal distribution is a continuous distribution so you do not need to worry about whether the inequality is strict (< or >) or weak (≤ or ≥)
• A value of z = 1 corresponds with the x-value that is 1 standard deviation above the mean and a value of z = -1 corresponds with the x-value that is 1 standard deviation below the mean
• If a value of x is less than the mean then the z -value will be negative
• The function is used to represent

#### How is the table of the normal distribution function used?

• In your formula booklet you have the table of the normal distribution which provides probabilities for the standard normal distribution
• The probabilities are provided for
• To find other probabilities you should use the symmetry property of the normal distribution curve
• The table gives probabilities for values of z between 0 and 4
• For negative values of z, the symmetry property of the normal distribution is used
• For values greater than z = 4 the probabilities are small enough to be considered negligible
• The tables give the probabilities to 4 decimal places
• To read probabilities from the normal distribution table for a z value :
• Find the z value in a column labelled z at the top
• The probability  will be the probability directly to the right of the z value
• So the value of  would be found to the right of 1.23
• If the value of z is not listed then find the closest value on the table
• z = 3.14 is not listed so use z = 3.15
• z = 2.25 is not listed so use z = 2.24 or z = 2.26

#### How is the table used to find probabilities that are not listed?

• The property that the area under the graph is 1 allows probabilities to be found for P( Z > z)
• Use the formula
• The symmetrical property of the normal distribution gives the following results:
• This allows probabilities to be found for negative values of z or for
• In summary
•
•
•
•
• Drawing a sketch of the normal distribution will help find equivalent probabilities
• Always consider whether the probability should be bigger or smaller then 0.5

#### How are z values found from the table of the normal distribution function?

• To find the value of z for which look for the value of p from within the table and find the corresponding value of z
• If the probability is given to 4 decimal places most of the time the value will exist somewhere in the tables
• If your probability is 0.5 or greater look through the tables to find the corresponding z value
• For   use the z value found in the table
• For take the negative of the z value found in the table
• If the probability is less than 0.5 you will need to subtract it from one before using the tables to find the corresponding z value
• For   take the negative of the z value found in the table
• For  use the z value found in the table
• Always draw a sketch so that you can see these clearly
• The formula booklet also contains a table of the critical values of z for
• This gives z values to 4 decimal places for common probabilities
• The probabilities in this table are 0.5, 0.4, 0.3, 0.2, 0.15, 0.1, 0.05, 0.025, 0.01, 0.005, 0.001 and 0.0005.

#### Worked example

(a)
By sketching a graph and using the table of the normal distribution, find the following:
(i)
(ii)
(iii)
(iv)

(b)
Find the value of such that
(a)
By sketching a graph and using the table of the normal distribution, find the following:
(i)
(ii)
(iii)
(iv)

(b)

Find the value of such that

#### Exam Tip

• Check whether the area shaded is more or less than 50% and compare this with your answer

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