# Standard Normal Distribution(CIE A Level Maths: Probability & Statistics 1)

Author

Amber

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 it is a good idea to learn how to use it to check your answers but you must still use the tables of the normal distribution and show all your working clearly
• 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 it is a normal 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 3
• For negative values of z, the symmetry property of the normal distribution is used
• For values greater than z = 3 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 of up to 2 decimal places:
• The very first column lists all z values to 1 decimal place from z = 0.0 to z = 2.9
• The top row gives the second decimal place for each of these z values
• So the value of  would be found at the point where the ‘1.2’ row meets the ‘3’ column
• To read probabilities from the normal distribution table for a z value of 3 decimal places:
• There is an extra section to the right of the tables that gives the amount to add on to the probabilities for the third decimal place
• The values given in the columns represent one ten-thousandth
• If the value is 7 we add 0.0007 to the probability
• If the value is 23 we add 0.0023 to the probability
• To find the value of  we would need to find the amount to add on to 0.8907
• Find the point where the 1.2 row meets the ADD 4 column, this gives us the number 7
• Add the value 0.0007 to the probability for

#### 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
•
• Therefore:
• The four cases in terms of  are:
• Drawing a sketch of the normal distribution will help find equivalent probabilities

#### 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
• Occasionally you may have to use the ADD columns to find the exact value
• If the values in the ADD columns don’t exactly match up use the closest value or find the midpoint of the z values that are either side of the probability
• 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
• This gives z values to 3 decimal places for common probabilities
• The probabilities in this table are 0.75, 0.9, 0.95, 0.975, 0.99, 0.995, 0.9975, 0.999 and 0.9995

#### 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

• A sketch will always help you to visualise the required probability and can be used to check your answer. Check whether the area shaded is more or less than 50% and compare this with your answer.

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### Author:Amber

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.