Standardisation of Normal Variables (DP IB Analysis & Approaches (AA)) : 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 

  • 

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

• Some mathematicians use the function  to represent 

z-values

What are z-values (standardised values)?

• For a normal distribution the z-value (standardised value) of an x-value tells you how many standard deviations it is away from the mean

  • If z = 1 then that means the x-value is 1 standard deviation bigger than the mean

  • If z = -1 then that means the x-value is 1 standard deviation smaller than the mean

• If the x-value is more than the mean then its corresponding z-value will be positive

• If the x-value is less than the mean then its corresponding z-value will be negative

• The z-value can be calculated using the formula:

  •  

  • This is given in the formula booklet

• z-values can be used to compare values from different distributions

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  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

  • or 

• 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 such that or 

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

  • Try to use lots of decimal places for the z-value or store your answer 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 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 x