The normal distribution

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When to use the normal distribution

The normal distribution models a continuous measurement that clusters symmetrically around an average — heights, weights, exam marks, times.
It is a bell-shaped curve described by its mean $\mu$ (centre) and standard deviation $\sigma$ (spread).
The total area under the curve is 1, and an area represents a probability.

Standardising with the z-score

To find probabilities you convert a value to a z-score — the number of standard deviations from the mean:

$z = \frac{x - \mu}{\sigma}$

Then a calculator (or standard normal table) turns the z-score into an area (probability). Positive $z$ is above the mean, negative below.

The inverse normal

An inverse problem gives a probability and asks for the value $x$ .
Work backwards: find the z-score for that area, then use $x = \mu + z\sigma$ .

Worked ExampleA normal probability

The masses of apples are normally distributed with mean $\mu = 170$ g and standard deviation $\sigma = 8$ g. Find the probability an apple weighs more than 180 g.

Step 1 — Standardise

$z = \frac{180 - 170}{8} = \frac{10}{8} = 1.25$

Step 2 — Sketch and identify the area

We want $P(X > 180)$ — the area to the right of $z = 1.25$ .

Step 3 — Read the probability

$P(Z > 1.25) = 1 - P(Z < 1.25) = 1 - 0.894 = 0.106$

So about 10.6% of apples weigh more than 180 g.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A normal distribution has mean 60 and standard deviation 5. Find the z-score of 72.

Merit

Heights are normally distributed with mean 170 cm and standard deviation 10 cm. Find the probability a person is shorter than 185 cm.

Excellence

Bags of sugar are normally distributed with mean 1000 g and standard deviation 12 g. The lightest 2.5% are rejected. Find the minimum acceptable weight.