The normal distribution and the empirical rule for HSC Maths Standard 2

What this dot point is asking

NESA wants three things. First, recognise the features of the normal distribution: it is symmetric, bell-shaped, and set by its mean and standard deviation. Second, apply the empirical $68$-$95$-$99.7$ rule to find the percentage of data inside standard-deviation bands. Third, use this to solve practical problems. This is one of the most reliably examined ideas in the Statistical Analysis module. The marks are easy once you stop thinking in raw numbers and start thinking in standard deviations.

The answer

The normal distribution

The normal distribution (or bell curve) is a smooth curve that shows how data is spread out. It turns up naturally whenever a quantity is built from many small, separate effects added together. Common examples are adult heights, exam scores across a large group, repeated measurement errors, and the fill weight of mass-produced bottles. Its shape is set entirely by two numbers:

• $\mu$ (mu): the mean, which sits under the peak and the centre of symmetry.
• $\sigma$ (sigma): the standard deviation, which controls the width. A small $\sigma$ gives a tall narrow curve; a large $\sigma$ gives a short wide one.

Here is the insight that makes every question easy to handle. Once you know $\mu$ and $\sigma$, you know the whole curve. Every normal curve is the same shape, just stretched or shifted. So a question never really cares about the raw value $x$. It only cares about how many standard deviations $x$ sits from the mean. That count is the only thing that decides a percentage.

Key properties to be able to state:

• Symmetric about the mean, so the left and right halves mirror each other.
• Mean $=$ median $=$ mode $= \mu$ (a consequence of the symmetry and single peak).
• Highest at $x = \mu$, falling away smoothly on both sides and never quite touching the axis.
• The total area under the curve is $1$, because it is a probability density: an area is a proportion of the data.

The empirical rule, band by band

The empirical rule is the main result here. Do not just memorise three separate numbers. Instead, build the rule up one band at a time, because that is exactly how the harder questions are set out step by step. The diagrams below shade one more standard-deviation band at each stage.

Stage 1, the central $68\%$. Go one standard deviation either side of the mean, from $\mu - \sigma$ to $\mu + \sigma$. About $68\%$ of all values fall in this central band. This is the bulk of the data, clustered near the mean.

Stage 2, out to $95\%$. Widen the band to two standard deviations either side, $\mu - 2\sigma$ to $\mu + 2\sigma$. Now about $95\%$ of values are captured. The extra strip on each side (from $1\sigma$ to $2\sigma$ out) adds $13.5\%$ per side, which is how $68\%$ grows to $95\%$.

Stage 3, out to $99.7\%$. Go three standard deviations either side, $\mu - 3\sigma$ to $\mu + 3\sigma$, and you have about $99.7\%$ of the data. Almost nothing lies beyond $3$ standard deviations: only $0.3\%$ in total, split between the two tails.

Stage 4, the half-band percentages. Because the curve is symmetric, each band splits evenly about the mean, so it is worth knowing the percentage of each individual strip. From the mean outward each side reads $34\%$, then $13.5\%$, then $2.35\%$, then $0.15\%$ in the far tail. These are the pieces you add and subtract to answer any "between these two values" question.

The tails, by symmetry

The single most useful follow-up move is splitting "inside the band" into "outside the band", then halving for one tail. If $68\%$ is inside $\mu \pm \sigma$, then $100 - 68 = 32\%$ is outside, and by symmetry each tail carries half:

• Above $\mu + \sigma$ (or below $\mu - \sigma$): $\dfrac{100 - 68}{2} = 16\%$.
• Above $\mu + 2\sigma$ (or below $\mu - 2\sigma$): $\dfrac{100 - 95}{2} = 2.5\%$.
• Above $\mu + 3\sigma$ (or below $\mu - 3\sigma$): $\dfrac{100 - 99.7}{2} = 0.15\%$.

Common standard-deviation regions

This table is the empirical rule in its most usable form. Every "what percentage" question is built by adding or subtracting these strips.

As a check, the strips on one side of the mean add to a half: $34 + 13.5 + 2.35 + 0.15 = 50\%$.

Applying the rule: the three-step method

To find the percentage of data in a range:

1. Express each endpoint as a number of standard deviations from the mean, using $\dfrac{\text{value} - \mu}{\sigma}$. You are turning raw values into the only currency that matters.
2. Sketch the curve, mark the mean and the two endpoints, and shade the region you want.
3. Add or subtract the strips from the region table to total the shaded area.

The sketch is not optional flourish: it stops the most common error, which is forgetting which strips lie inside the region you were actually asked about.

If the endpoints are not whole numbers of standard deviations from the mean, the empirical rule cannot give an exact answer. Standard 2 then expects z-scores and the standard normal table (the next dot point); the empirical rule is the special case where the endpoints land exactly on $\pm 1$, $\pm 2$ or $\pm 3$.

When the normal distribution applies

The rule only works when the data is actually (approximately) normal. The model fits when:

• Natural variation is at play: adult heights and weights, exam scores in a large cohort, IQ scores.
• Measurement error accumulates: repeated readings of the same fixed quantity.
• Manufacturing tolerances apply: fill weights, component dimensions on a production line.

It does not fit obviously skewed data (household incomes, house prices, reaction times), where a few extreme values pull one tail out. Applying the $68$-$95$-$99.7$ percentages to skewed data is a conceptual error that costs marks.

How exam questions ask about the normal distribution

The wording changes but the task is always "turn endpoints into standard deviations, then add strips". Learn the translations:

• "What percentage lie between $a$ and $b$?" Convert both endpoints to standard deviations from the mean, then sum the strips between them.
• "What percentage are more than / greater than $a$?" A tail question. Find how many SDs $a$ is above the mean, take the inside-band percentage, subtract from $100$, then halve for the single tail (or read the tail directly: $16\%$, $2.5\%$, $0.15\%$).
• "What percentage are less than $a$?" Everything to the left: $50\%$ for the whole left half, plus or minus the strips between the mean and $a$.
• "How many students / bottles / people ...?" Find the percentage, then multiply by the total. Give a whole number for a count of people.
• "Between what two values do the middle $95\%$ lie?" Run the rule backwards: the middle $95\%$ is $\mu \pm 2\sigma$, so compute those two values.
• "Is it unusual to score above $x$?" Code for "how far into the tail is it": a value beyond $2\sigma$ (top $2.5\%$) is uncommon; beyond $3\sigma$ (top $0.15\%$) is rare.