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

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?

• 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

• Some mathematicians use the function   to represent 

The table of percentage points of the normal distribution

• In your formula booklet you have the table of percentage points which provides information about specific values of the standard normal distribution that correspond to commonly used probabilities

  • 

  • You are given the value of to 4 decimal places when p  is:

    • 0.5, 0.4, 0.3, 0.2, 0.15, 0.1, 0.05, 0.025, 0.01, 0.005, 0.001, 0.005

• These values of z can be found using the "Inverse Normal Distribution" function on your calculator

  • If you are happy using your calculator then you can simply ignore this table

• They are simply listed in your formula booklet as they are commonly used when:

  • Finding an unknown mean and/or variance for a normal distribution

  • Performing a hypothesis test on the mean of a normal distribution

Finding Sigma and Mu

How do I find the mean (μ) or the standard deviation (σ) if one of them is unknown?

• 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

How do I find the mean (μ) and the standard deviation (σ) if both of them are unknown?

• 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