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The normal distribution and Z

The normal distribution is bell-shaped about the mean. Standardising subtracts the mean and divides by the standard deviation, so you get a Z-value to look up. One standard deviation below and above the mean is an interval the course links to about 68% of the observations. The table look-up itself, for example that Φ of 1 is about 0.841, is a table value and not an arithmetic calculation with the four operations.

Try it yourself

The normal distribution

Z = \frac{X - \mu}{\sigma}

standardising

\mu \pm \sigma

one standard deviation out

Symbols

$\mu$	mean
$\sigma$	standard deviation
$Z$	standardised value

Example

Height $N(175, 7)$ cm. $Z$ for 182 cm is $\dfrac{182 - 175}{7} = 1$ .

The interval $\mu \pm \sigma$ runs from $175 - 7 = 168$ cm to $175 + 7 = 182$ cm.

Calculate Z. You read the proportion in the normal table.

Practise probability distributions for free →

← The binomial distribution · A test of a proportion →

Part of Mathematics S2: Probability distributions.