StatisticsIntroductory StatisticsHigh school

The Normal Distribution

z-scores and areas under the normal curve.

The normal distribution is the symmetric bell-shaped curve that describes many measurements, from heights to test scores to measurement errors. It is completely determined by its mean $\mu$ and standard deviation $\sigma$ . About $68\%$ of values fall within one $\sigma$ of the mean, $95\%$ within two and $99.7\%$ within three.

A z-score $z = \frac{x - \mu}{\sigma}$ counts how many standard deviations a value sits from the mean, which lets you compare values from different normal distributions and find probabilities from a single standard normal table.

Key terms

z-score: How many standard deviations a value is from the mean.
$z = \frac{x - \mu}{\sigma}$

Key ideas

Standardise: $z = \frac{x - \mu}{\sigma}$ ; unstandardise: $x = \mu + z\sigma$ .
The empirical rule: $68$ - $95$ - $99.7$ .
Probabilities are areas under the curve; total area is 1.
By the central limit theorem, sample means are approximately normal for large samples.

Worked example

Test scores are normal with $\mu = 70$ and $\sigma = 8$ . What share of students score above 86?

$z = \frac{86 - 70}{8} = 2$ .
About $95\%$ lie within $z = \pm 2$ , leaving $5\%$ in the two tails.
By symmetry, half of that is above $z = 2$ .
About $2.5\%$ (the exact table value is $2.28\%$ ).

Answer: about $2.3\%$

Common mistakes

Forgetting to divide by $\sigma$ when standardising.
Using the empirical rule for data that are clearly skewed.
Reporting the area to the left when the question asks for the right.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.Find $z$ for $x = 130$ when $\mu = 100$ and $\sigma = 15$ .
Show answer
$2$
2.Which value has $z = -1.5$ when $\mu = 50$ and $\sigma = 4$ ?
Show answer
$44$
3.Roughly what percent of a normal population lies within one standard deviation of the mean?
Show answer
$68\%$

Keep practising

Calculators

Z-Score Calculator