Standardisation of Normal Variables & z-Values (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 

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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:

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  • Where and 

• Probabilities are related by:

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  • 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:

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  • This is given in the formula booklet

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