Standard Normal Distribution (AQA A Level Maths): Revision Note

The standard normal distribution is a normal distribution where the mean is 0 and the standard deviation is 1

• It is denoted by Z

• 

Any normal distribution curve can be transformed to the standard normal distribution curve by a horizontal translation and a horizontal stretch

Therefore we have the relationship:

• 

• Where  and 

Probabilities are related by:

•  

• This will be useful when the mean or variance is unknown

If a value of x is less than the mean then the z-value will be negative

In old textbooks you might see the function this just means

If the mean or standard deviation of the  is unknown then you will need to use the standard normal distribution

You will need to use the formula

• or its rearranged form 

You will be given a probability for a specific value of  

To find the unknown parameter:

STEP 1: Sketch the normal curve

• Label the known value and the mean

STEP 2: Find the z-value for the given value of x

• Use the Inverse Normal Distribution to find the value of z such that  or 

• Make sure the direction of the inequality for Z is consistent with X

• Try to use lots of decimal places for the z-value to avoid rounding errors

  • You should use at least one extra decimal place within your working than your intended degree of accuracy for your answer

STEP 3: Substitute the known values into or 

• You will be given x and one of the parameters (μ  or σ) in the question

• You will have calculated z in STEP 2

STEP 4: Solve the equation

If both of them are unknown then you will be given two probabilities for two specific values of x

The process is the same as above

• You will now be able to calculate two z-values

• You can form two equations (rearranging to the form  is helpful)

• You now have to solve the two equations simultaneously (you can use your calculator to do this)

• Be careful not to mix up which z-value goes with which value of