Standardisation of Normal Variables & z-Values (DP IB Analysis & Approaches (AA): HL): 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 $Z$

  • $Z~\text{N}(0,1^2)$

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 combined horizontal and vertical stretch

• The relationship between the variables is given by:

  • $Z=\frac{X-\mu }{\sigma }$ 

  • Where $X~\text{N}(\mu ,{\sigma }^2)$ and $Z~\text{N}(0,1^2)$

• Probabilities are related by:

  • $\mathrm{P}(a<X<b)=\mathrm{P}(\frac{a-\mu }{\sigma }<Z<\frac{b-\mu }{\sigma })$ 

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

• Some mathematicians use the function notation $\mathrm{\Phi }\left(z\right)$ to represent $\mathrm{P}(Z<z)$

z-values

What are z-values (standardised values)?

• For a normal distribution $X~\mathrm{N}(\mu ,{\sigma }^2)$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 an x value is more than the mean then its corresponding z-value will be positive

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

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

  • $z=\frac{x-\mu }{\sigma }$ 

  • This is given in the formula booklet

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