Recognise the features of the normal distribution and apply the empirical $68$-$95$-$99.7$ rule

What this dot point is asking

NESA wants you to recognise the features of the normal distribution (symmetric, bell-shaped, characterised by mean and standard deviation), apply the empirical $68$-$95$-$99.7$ rule to find percentages of data within standard-deviation bands, and use this to solve practical problems.

The answer

The normal distribution

The normal distribution (or bell curve) is a continuous probability distribution that arises naturally in many contexts (heights, exam scores, measurement errors, manufacturing variation). Its shape is fully specified by two parameters:

• IMATH_6 (mu): the mean.
• IMATH_7 (sigma): the standard deviation.

Key properties:

• Symmetric about the mean.
• Mean = median = mode = $\mu$.
• Highest at $x = \mu$, then falls off on both sides.
• Total area under the curve is $1$ (it is a probability density).
• Extends from $-\infty$ to $+\infty$ (in principle).

The empirical rule (68-95-99.7)

For any normal distribution:

• About $68\%$ of values lie within $1$ standard deviation of the mean: $\mu \pm \sigma$.
• About $95\%$ lie within $2$ standard deviations: $\mu \pm 2\sigma$.
• About $99.7\%$ lie within $3$ standard deviations: $\mu \pm 3\sigma$.

By symmetry, the tails (one side) are half the outside-the-band amount:

• Above $\mu + \sigma$: $\frac{100 - 68}{2} = 16\%$.
• Above $\mu + 2\sigma$: $\frac{100 - 95}{2} = 2.5\%$.
• Above $\mu + 3\sigma$: $\frac{100 - 99.7}{2} = 0.15\%$.

Common standard-deviation regions

These add up: $0.15 + 2.35 + 13.5 + 34 = 50\%$ from the mean to the right tail.

Applying the rule

To find the percentage of data in some range:

1. Express each endpoint in terms of standard deviations from the mean.
2. Use the empirical rule values or the region table.
3. Sum or subtract regions as needed.

If endpoints are not at exact integer standard deviations from the mean, Standard 2 expects you to use z-scores and a calculator (covered in the next dot point).

When the normal distribution applies

• Natural variation. Adult heights, weights, exam scores in large populations, IQ scores.
• Measurement error. Repeated measurements of the same quantity.
• Manufacturing. Weights of mass-produced items, dimensions of components.
• Sums and averages. By the Central Limit Theorem (covered in Maths Advanced), means of large samples are approximately normal regardless of the original distribution.

The normal distribution is the default model whenever a quantity is the result of many small, independent influences adding up.