The normal distribution (SACE Stage 2 Mathematical Methods)

Syllabus dot point

“The normal distribution is a symmetric bell-shaped density defined by its mean and standard deviation, with predictable probabilities given by the empirical rule.”

8 min readUpdated 2 Jun 2026

Quick answer

The normal distribution is a symmetric, bell-shaped continuous distribution defined by its mean $\mu$ (centre) and standard deviation $\sigma$ (spread), written $X\sim N(\mu,\sigma^2)$ . About $68\%$ of values lie within $1\sigma$ of the mean, $95\%$ within $2\sigma$ , and $99.7\%$ within $3\sigma$ - the empirical rule.

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What this dot point is asking
Properties of the bell curve
The empirical rule (68-95-99.7)
Why so many quantities are normal
Comparing two normal distributions
Common errors
Why it matters
Exam-style questions

What this dot point is asking

The normal distribution is the most important continuous distribution in statistics. Many natural and measured quantities - heights, exam scores, measurement errors - follow it closely, and it is the limiting shape of the binomial distribution for large $n$ .

Properties of the bell curve

Features of the normal distribution

Symmetric about the mean $\mu$ ; the mean, median and mode all coincide there.
Bell-shaped, with total area under the curve equal to $1$ .
The mean $\mu$ locates the centre; the standard deviation $\sigma$ controls the width (large $\sigma$ = flatter and wider, small $\sigma$ = taller and narrower).
Asymptotic: the tails approach but never touch the x-axis.

The notation $X\sim N(\mu,\sigma^2)$ means $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$ (so standard deviation $\sigma$ ).

The empirical rule (68-95-99.7)

For any normal distribution, fixed proportions of the data fall within whole numbers of standard deviations of the mean:

The 68-95-99.7 rule

P(\mu-\sigma\le X\le \mu+\sigma)\approx 0.68

P(\mu-2\sigma\le X\le \mu+2\sigma)\approx 0.95

P(\mu-3\sigma\le X\le \mu+3\sigma)\approx 0.997

Because the curve is symmetric, each tail beyond $\pm 1\sigma$ holds about $16\%$ , beyond $\pm 2\sigma$ about $2.5\%$ , and beyond $\pm 3\sigma$ about $0.15\%$ .

Applying the empirical rule

Heights in a population are $N(170,\,8^2)$ cm, meaning a mean of $170$ and $\sigma=8$ . This example estimates the proportion of people taller than $186$ cm.

Step 1: express 186 cm in terms of $\mu$ and $\sigma$

To use the empirical rule, we first check how many standard deviations $186$ cm is from the mean. Computing $186 - 170 = 16 = 2\times 8$ , so $186 = \mu + 2\sigma$ . The value sits exactly two standard deviations above the mean.

Step 2: apply the empirical rule to find the tail proportion

The empirical rule tells us that $95\%$ of values lie within $2\sigma$ of the mean, leaving $5\%$ in the two tails combined. Because the normal distribution is symmetric, each tail holds equal probability, so the proportion above $\mu + 2\sigma$ is half of $5\%$ :

P(X>186)\approx \frac{5\%}{2}=2.5\%.

Final answer: approximately $2.5\%$ of the population are taller than $186$ cm.

Proportion within a range

For the same heights $N(170,8^2)$ , this example estimates the proportion of people with height between $162$ cm and $178$ cm.

Step 1: identify both endpoints as $\mu \pm \sigma$

Before applying any rule, we express both endpoints in terms of the mean and standard deviation. Here $162 = 170 - 8 = \mu - \sigma$ and $178 = 170 + 8 = \mu + \sigma$ , so the interval from $162$ to $178$ is exactly the region within one standard deviation of the mean.

Step 2: apply the 68% rule

The empirical rule states directly that approximately $68\%$ of values in any normal distribution lie within one standard deviation of the mean:

P(162\le X\le 178)\approx 0.68=68\%.

Final answer: approximately $68\%$ of the population have heights between $162$ cm and $178$ cm.

Why so many quantities are normal

The normal distribution appears everywhere because of the central limit theorem, which you meet in Topic 6: when many small independent effects add together, their sum tends toward a normal shape regardless of the individual distributions. Human heights, measurement errors and manufacturing variations all arise this way, as the accumulation of many tiny influences. This is also why the binomial distribution, which counts the sum of many independent Bernoulli trials, looks more and more normal as $n$ grows. Recognising the underlying additive structure is what justifies modelling a quantity as normal in the first place.

Comparing two normal distributions

Two normal distributions with the same mean but different standard deviations have their peaks at the same place, but the one with the larger $\sigma$ is shorter and wider because the fixed total area of $1$ must spread over a greater range. Two distributions with the same $\sigma$ but different means are identical in shape, just shifted along the axis. SACE diagrams often show two bell curves and ask you to compare their means and spreads; read the centre for the mean and the width (or the height of the peak) for the standard deviation.

Common errors

Confusing variance with standard deviation in N(μ, σ²)

The notation $N(\mu,\sigma^2)$ states the variance as the second parameter, but the empirical rule and z-scores use the standard deviation $\sigma$ . If you read $N(170,64)$ , the standard deviation is $\sqrt{64}=8$ , not $64$ . Always check whether you have been given $\sigma$ or $\sigma^2$ before applying the $68$ - $95$ - $99.7$ distances.

Mark the standard-deviation gridlines first

When sketching, draw and label the vertical lines at $\mu$ , $\mu\pm\sigma$ , $\mu\pm 2\sigma$ , $\mu\pm 3\sigma$ before shading. This turns most empirical-rule questions into reading off the right slice (68, 95 or 99.7, halved for one-sided tails).

Why it matters

The normal distribution underpins the rest of Topic 5 (z-scores and exact probabilities) and all of Topic 6 (sampling distributions and confidence intervals), where sample means are approximately normal. Recognising when data is normal and using the empirical rule for quick estimates is a core Stage 2 skill.

Exam-style questions

Questions in the style of SACE Board exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.

SACE 2024-style2 marks

Calculator-assumed. Let

X

be normally distributed with mean

54

and standard deviation

6

. Determine (a)

\Pr(50 < X < 52)

and (b)

\Pr(X > 58)

Show worked answer →

Use the normal distribution with $\mu = 54$ , $\sigma = 6$ (calculator normal CDF, or standardise with $z = \dfrac{x - 54}{6}$ ).

(a) $z$ ranges from $\dfrac{50 - 54}{6} = -0.667$ to $\dfrac{52 - 54}{6} = -0.333$ , so $\Pr(-0.667 < Z < -0.333) \approx 0.117$ . (1 mark)

(b) $z = \dfrac{58 - 54}{6} = 0.667$ , so $\Pr(Z > 0.667) = 1 - 0.7475 \approx 0.252$ . (1 mark)

Calculator values to three significant figures are accepted. Both regions lie within one standard deviation of the mean, so neither probability is extreme.

SACE 2023-style1 mark

Calculator-assumed. The daily milk production of a randomly chosen cow is normally distributed with mean

21.9

litres and standard deviation

3.26

litres. Determine the probability that production is less than

23

litres.

Show worked answer →

Standardise: $z = \dfrac{23 - 21.9}{3.26} = \dfrac{1.1}{3.26} \approx 0.337$ .

Then $\Pr(X < 23) = \Pr(Z < 0.337) \approx 0.632$ , from the standard normal or the calculator's normal CDF.

So the probability is $0.632$ . Because $23$ is just above the mean $21.9$ , the answer is a little more than $0.5$ , which checks out.

SACE 2022-style2 marks

Calculator-free. The masses of apples are

N(150, 20^2)

grams. Using the empirical rule, estimate (a) the proportion of apples between

130

g and

170

g, and (b) the proportion heavier than

190

Show worked answer →

(a) $130 = \mu - \sigma$ and $170 = \mu + \sigma$ , so this is within one standard deviation: about $68\%$ . (1 mark)

(b) $190 = \mu + 2\sigma$ . The empirical rule gives $95\%$ within $2\sigma$ , so $5\%$ in the two tails, and by symmetry the upper tail holds half: about $2.5\%$ . (1 mark)

Marks reward recognising the values as whole numbers of standard deviations from the mean and applying the $68$ - $95$ - $99.7$ rule with the tail halving.

Practise this

Sources & how we know this

Stage 2 Mathematical Methods subject outline (2025 onwards) — SACE Board
Stage 2 Mathematical Methods: external assessment — SACE Board

What this dot point is asking

Properties of the bell curve

The empirical rule (68-95-99.7)

Step 1: express 186 cm in terms of μ\muμ and σ\sigmaσ

Step 2: apply the empirical rule to find the tail proportion

Step 1: identify both endpoints as μ±σ\mu \pm \sigmaμ±σ

Step 2: apply the 68% rule

Why so many quantities are normal

Comparing two normal distributions

Common errors

Why it matters

Exam-style questions

Practise this

Sources & how we know this

Step 1: express 186 cm in terms of $\mu$ and $\sigma$

Step 1: identify both endpoints as $\mu \pm \sigma$