The normal distribution - CCEA GCSE Statistics guide to the bell curve, the 68 to 95 to 99.7 rule and standardised scores

The normal distribution is the Higher-tier model for naturally varying continuous data. This CCEA GCSE Statistics guide covers its properties, the 68 to 95 to 99.7 rule, and how standardised scores find proportions.

Properties of the normal distribution

The normal distribution is a symmetrical bell-shaped curve centred on the mean, and it models much real data such as heights, masses and exam marks. Its mean, median and mode are all equal and lie at the centre, the curve is symmetrical about the mean, most data clusters near the mean, and the tails thin out the further you go. The standard deviation controls the spread, so a larger standard deviation gives a wider, flatter curve. The total area under the curve represents all the data. A quick check that data may be normal is that the mean and median are roughly equal, since clearly skewed data is not normal.

The 68 to 95 to 99.7 rule

The rule that powers the calculations is how data is spread around the mean. About 68 percent of the data lies within one standard deviation of the mean, about 95 percent within two standard deviations, and about 99.7 percent within three. Because the curve is symmetrical, half of each percentage lies on either side of the mean, and the small amount left in the tails is split equally. The skill is to express the given values as a number of standard deviations from the mean, apply the rule, and use the symmetry to find tail proportions.

Using symmetry to find proportions

To find a proportion, split the interval at the mean and treat each side separately, because the curve is symmetrical. Within one standard deviation below the mean is half of 68 percent, so 34 percent; within two standard deviations above is half of 95 percent, so 47.5 percent; and beyond two standard deviations above lies half of the leftover 5 percent, so 2.5 percent. Adding the relevant parts gives the proportion in any interval expressed in whole numbers of standard deviations.

Standardised scores

A standardised score says how many standard deviations a value is from the mean, so scores of plus or minus one, two and three line up exactly with the 68 to 95 to 99.7 rule. A value with a standardised score of two sits where only about 2.5 percent of the data lies beyond it. Standardised scores therefore let you find proportions in a normal distribution and compare values from different normal distributions, linking this topic to the averages and spread module.

How CCEA examines the normal distribution

CCEA rewards stating the properties of the normal distribution, using the 68 to 95 to 99.7 rule with the symmetry of the curve to find percentages and proportions, recognising when data is approximately normal, and connecting the rule to standardised scores. Use the dot point for specification-level detail and worked CCEA-style questions, then test yourself with the quiz.