HSC Mathematics Standard 2 the normal distribution (2026 guide)

Figure 1. The standard normal distribution and the empirical 68-95-99.7 rule. The variable scale (top labels) and the standard-score $z$-scale (bottom labels) are aligned: any value $x$ converts to a $z$-score via $z = (x - \mu)/\sigma$. Approximately 68% of the probability mass lies within $\pm 1\sigma$, 95% within $\pm 2\sigma$, and 99.7% within $\pm 3\sigma$ of the mean.

The normal distribution is one of the cornerstone topics of HSC Mathematics Standard 2 statistics. Every paper since the 2017 syllabus reform has had at least one question on z-scores, and most have a separate question on the empirical rule. Across the paper, you can expect about $8$-$12$ marks from this topic.

The good news: the normal distribution is conceptually one of the cleanest topics in the syllabus. The 68-95-99.7 rule is genuinely memorisable, the z-score formula is on the reference sheet, and the calculator does the heavy lifting for non-standard probabilities. The bad news: students lose easy marks every year through sign errors, confusing standard deviation with variance, and applying the rule to non-normal data.

What the normal distribution is

The normal distribution (or bell curve) is a continuous probability distribution fully specified by two parameters:

• $\mu$ (mu): the mean.
• $\sigma$ (sigma): the standard deviation.

Notation: $X \sim N(\mu, \sigma^2)$. Note the variance, not the standard deviation, sits inside.

Properties:

• Symmetric about $x = \mu$.
• Mean = median = mode = $\mu$.
• Total area under the curve is $1$.
• Higher $\sigma$ gives a flatter, wider curve. Lower $\sigma$ gives a taller, narrower curve.

The bell shape arises naturally in many contexts, especially when the quantity is the sum of many small independent random influences (heights, exam marks in large cohorts, measurement errors, manufacturing variation).

The empirical rule

For any normal distribution:

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

By symmetry, the tail percentages above the upper threshold are half the percentage outside the band:

• Above $\mu + \sigma$: $16\%$.
• Above $\mu + 2\sigma$: $2.5\%$.
• Above $\mu + 3\sigma$: $0.15\%$.

Standard region percentages:

Memorise this table; it is the heart of empirical-rule questions.

z-scores

The z-score of a value $x$ from a normal distribution with mean $\mu$ and standard deviation $\sigma$:

The z-score is the number of standard deviations $x$ is from the mean.

To find $x$ given $z$:

Common percentile-to-z-score values

By symmetry, lower percentiles use the negative of the upper: 10th = $-1.28$, etc.

Using the standard normal table

The HSC Standard 2 paper provides a table of $\Phi(z) = P(Z \le z)$, the cumulative distribution function of the standard normal.

For any value $x$ from a normal distribution:

• $P(X \le x) = \Phi(z)$ where $z = (x - \mu) / \sigma$.
• $P(X > x) = 1 - \Phi(z)$.
• $P(a \le X \le b) = \Phi(z_b) - \Phi(z_a)$.

Australian context examples

Heights from the ABS

ABS data (Australian Health Survey, 2017-18) gives mean adult male height around $175$ cm with standard deviation about $7$ cm. The empirical rule then says:

• $68\%$ of men are between $168$ and $182$ cm.
• $95\%$ are between $161$ and $189$ cm.
• $2.5\%$ are taller than $189$ cm.

HSC marks distribution

HSC exam marks are scaled, but a rough approximation is that raw marks in many subjects are approximately normally distributed with mean about $65$-$70$ and standard deviation about $12$-$15$. The empirical rule then gives a rough sense of how rare Band 6 (typically the top $\sim 10\%$) is in raw terms.

For a quick estimate using $\mu = 70$, $\sigma = 13$: the top $10\%$ requires $z \ge 1.28$, so $x \ge 70 + 1.28 \times 13 \approx 86.6$ raw marks for a Band 6 (matching the inverse-normal worked example above).

Common normal-distribution mistakes

Confusing $\sigma$ with $\sigma^2$

If $X \sim N(100, 25)$, then $\sigma = 5$ (the variance is $25$, the standard deviation is the square root).

Sign error in z-score

$z = (x - \mu) / \sigma$. A value below the mean has a negative z-score.

Forgetting to halve

"Within $\pm 1\sigma$" is $68\%$. "Above $\mu + \sigma$" is $16\%$ (half of $32\%$). Easy to halve the wrong number.

Using the rule on non-normal data

The empirical rule applies only to (approximately) normal data. Income, for example, is highly right-skewed; the rule does not apply.

Reading the wrong percentage from the table

$\Phi(z)$ is the area to the LEFT of $z$. For "greater than", use $1 - \Phi(z)$.

Exam strategy

For normal distribution questions in the HSC:

1. Always state the parameters. "$X \sim N(70, 13^2)$" or "mean $70$, SD $13$".
2. Show the z-score calculation explicitly. Even if the question is solvable by the empirical rule, marking guides usually accept either approach.
3. Sketch a normal curve. Shade the region in question. This catches errors about which tail you want.
4. State the final probability with units or percentage. "$P(X > 85) \approx 0.067$" or "$\approx 6.7\%$".

In the past three years of HSC papers, normal distribution and z-score questions have appeared in every paper, contributing $6$-$10$ marks. They are among the most reliable mark-earners in the paper for students who have drilled the empirical rule and the z-score formula.

Check your knowledge

Try these before checking the worked solutions. They cover empirical-rule reading, z-score computation, table lookup and inverse normal.

1. For $X \sim N(50, 8^2)$, what is the z-score of $x = 66$?
2. For $X \sim N(50, 8^2)$, give the value of $x$ whose z-score is $-1.25$.
3. In a normal distribution, approximately what percentage of values lie between $\mu - 2\sigma$ and $\mu + 1\sigma$?
4. Marks on a test are $N(60, 10^2)$. Use the empirical rule to estimate the percentage of students who scored above $80$.
5. For $Z \sim N(0, 1)$ and $\Phi(1.2) \approx 0.8849$, find $P(Z > 1.2)$.
6. Adult female heights in Australia are approximately $N(162, 7^2)$ (cm). Using $\Phi(1.43) \approx 0.9236$, find the proportion of women shorter than $172$ cm.
7. Two students sit different tests. Cara scored $74$ on a test with $\mu = 65$, $\sigma = 9$. Dan scored $80$ on a test with $\mu = 72$, $\sigma = 10$. Whose result was better, and by how many standard deviations?
8. ATAR-style raw marks on a paper are $N(68, 12^2)$. Find the cut-off raw mark above which the top $5\%$ of students sit. Use $z_{0.95} \approx 1.645$.