The normal distribution
- Many measurements, such as heights and exam marks, follow a normal distribution: a symmetrical, bell-shaped curve.
- The mean, median and mode are equal and lie at the centre.
- About 68% of the data lies within 1 standard deviation of the mean, about 95% within 2 standard deviations, and almost all (about 99.7%) within 3 standard deviations.
- Example: IQ scores with mean 100 and standard deviation 15. About 95% lie between 70 and 130.
Standardised scores
- A standardised score (z-score) shows how many standard deviations a value is from the mean: z = (value − mean) ÷ standard deviation.
- Example: mean 60, standard deviation 8, value 76. z = 16 ÷ 8 = 2.
- Standardised scores compare values from different distributions. Maths 70 (mean 60, SD 5) gives z = 2; English 75 (mean 65, SD 10) gives z = 1. So the maths result was relatively better.
- A negative z-score means the value is below the mean.
Quality control
- Quality assurance uses control charts. Samples are taken regularly and their means plotted.
- Warning limits are usually 2 standard deviations from the target mean, and action limits 3 standard deviations away. A point beyond an action limit means the process should be stopped and checked.
Key terms
- Normal distribution
- A symmetrical, bell-shaped distribution.
- Standardised score
- The number of standard deviations a value is from the mean.
- Control chart
- A chart used to check that a process stays on target.
- Warning limits
- Lines 2 standard deviations from the target mean on a control chart.
- Action limits
- Lines 3 standard deviations from the target mean on a control chart.
- Bell-shaped curve
- The shape of a normal distribution.
Practise The normal distribution and standardised scores: 10 questions