## 3.5 Finding Normal Distribution Probabilities

#### Standardised Values and Finding Normal Distribution Probabilities

- Previously, we see that any normal distribution can be converted into a standard one via

\operatorname{Pr}(X \leq a)=\operatorname{Pr}\left(Z \leq \frac{a-\mu}{\sigma}\right) \equiv \operatorname{Pr}(Z \leq z)

where X \sim N(\mu, \sigma), and Z \sim N(0,1).

- Such expression is useful as we convert any values into one that is in terms of standard deviation(s) away from the mean. These new values are called standardised values or z-values. In particular we have

z=\frac{x-\mu}{\sigma} \text { or standardised value }=\frac{\text { data value }-\text { mean of normal curve }}{\mathrm{s} . \mathrm{d} . \text { of the curve }}

- Therefore, we can infer that: A positive z-value indicates that the data value it represents lies above the mean. If it is negative, then it is below the mean.

- Knowing how to convert any normal distribution to a standard one is important as it helps us to find the probabilities of any normal distributed events.