The shape of the normal distribution, symmetry about the mean, and the 68, 95, 99.7 percent rule.

What this dot point is asking

AQA wants you to recognise the shape of the normal distribution, know it is symmetrical about the mean, understand how the standard deviation controls its width, and use the $68$, $95$, $99.7$ percent rule to estimate the proportion of data in a given range.

The shape of the normal distribution

Because it is symmetrical, the mean, median and mode all lie at the centre, directly under the peak. The curve never quite touches the horizontal axis: in principle values can lie any distance from the mean, but the chance of extreme values becomes vanishingly small. The normal distribution arises whenever a quantity is the result of many small, independent influences adding together, which is why heights, manufacturing variation and measurement errors so often look normal.

The role of the standard deviation

Two normal distributions can share a mean but look quite different: a sample of carefully machined parts has a small standard deviation (a tall, thin curve), while a sample of handmade parts has a large one (a short, wide curve). Total area under any normal curve is $1$ (it is a probability), so a taller curve must be narrower to keep the area fixed.

The 68, 95, 99.7 percent rule

By symmetry, the data outside a band is split equally between the two tails. Outside $\pm 1\sigma$ is $32\%$, so $16\%$ lies in each tail; outside $\pm 2\sigma$ is $5\%$, so $2.5\%$ in each tail; outside $\pm 3\sigma$ is $0.3\%$, so $0.15\%$ in each tail. Splitting the band from the mean outward gives the useful half-band figures: mean to $1\sigma$ is $34\%$, $1\sigma$ to $2\sigma$ is $13.5\%$, and $2\sigma$ to $3\sigma$ is $2.35\%$.

A reliable method for any range question is to convert each endpoint into a number of standard deviations from the mean, mark the half-band percentages on a quick sketch of the curve, and then add or subtract the regions you need. The exact figures are slightly higher than the round numbers (the bands are really $68.27\%$, $95.45\%$ and $99.73\%$), but the rounded $68$, $95$, $99.7$ values are what AQA expects. Note that the rule only gives percentages at whole numbers of standard deviations; for endpoints in between, such as $1.4\sigma$, you would need a standardised score and a table, which is the standardised-scores topic.