The normal distribution with mean $\mu$ and standard deviation $\sigma$, the standard normal $Z$, the use of the empirical 68/95/99.7 rule, and computation of normal probabilities and inverse probabilities using technology or standard tables

What this dot point is asking

VCAA wants you to recognise the normal distribution, apply the empirical 68/95/99.7 rule, standardise to the standard normal $Z$, and compute normal probabilities and inverse probabilities by hand (for Paper 1 empirical-rule questions) and by calculator (for Paper 2). The dot point is the bridge between the abstract pdf concept and applied probability questions.

The normal distribution

A continuous random variable $X$ is normally distributed if its pdf is:

Notation: $X \sim N(\mu, \sigma^2)$.

Properties:

• Symmetric about the mean. The pdf is bell-shaped, with peak at $x = \mu$.
• Mean = median = mode = $\mu$.
• Standard deviation = $\sigma$. Two-thirds of probability mass within $\pm \sigma$ of $\mu$.
• Support is $(-\infty, \infty)$. $f$ is positive but extremely small far from $\mu$.

The pdf itself is rarely computed by hand in VCE Methods; it is provided in the formula sheet and integrated by technology.

The empirical 68/95/99.7 rule

For any normal distribution:

• IMATH_14
• IMATH_15
• IMATH_16

Equivalently, in $Z$:

• IMATH_18
• IMATH_19
• IMATH_20

These are the values expected by hand on Paper 1. Use them whenever the question's $x$-values land neatly at $\mu \pm n \sigma$ for $n = 1, 2, 3$.

By symmetry, the probabilities in each tail beyond $\mu + n \sigma$ are:

• Beyond $\mu + \sigma$: $\frac{1 - 0.68}{2} = 0.16$
• Beyond $\mu + 2\sigma$: $\frac{1 - 0.95}{2} = 0.025$
• Beyond $\mu + 3\sigma$: IMATH_30

Standardisation: IMATH_31

The standard normal $Z \sim N(0, 1)$ has mean 0 and standard deviation 1. Any normal random variable $X$ can be converted to a standard normal by the transformation:

This is called standardisation. The transformation preserves probabilities:

The standardisation is the key skill: it converts any normal probability question to a question about $Z$, which has tabulated values and is supported on every CAS.

Worked standardisation

$X \sim N(50, 4^2)$. Find $P(X \leq 56)$.

$z = \frac{56 - 50}{4} = 1.5$.

$P(X \leq 56) = P(Z \leq 1.5) \approx 0.9332$ (from table or calculator).

Inverse normal: from probability to $x$-value

Given a probability $p$, find the $x$-value $c$ such that $P(X \leq c) = p$ (or $P(X > c) = 1 - p$).

Procedure:

1. Find $z$ such that $P(Z \leq z) = p$. Use the inverse-normal function on calculator: $z = \text{invNorm}(p)$, or use a table.
2. Convert back to $x$: $c = \mu + z \sigma$.

Common values:

These appear in confidence-interval questions (covered in the confidence-intervals dot point).

Worked inverse normal

$X \sim N(50, 0.4^2)$. Find $c$ such that $P(X > c) = 0.10$.

$P(X \leq c) = 0.90$. So $z = 1.2816$.

$c = 50 + 1.2816 \times 0.4 = 50.513$.

Calculator workflow (Paper 2)

For probability questions (find $P(a \leq X \leq b)$): use normCdf(a, b, mu, sigma) on TI-Nspire or equivalent on ClassPad.

For inverse questions (find $c$ given $P(X \leq c) = p$): use invNorm(p, mu, sigma).

State the set-up explicitly in the working: "$X \sim N(50, 0.4^2)$; $P(X \leq c) = 0.90$; from inverse-normal, $c \approx 50.513$."

Standard contexts

Quality control. The lengths or weights of manufactured items are often modelled as normal with mean = target and standard deviation = tolerance. Questions ask for the proportion outside specification.

Examinations and IQ. Scores on standardised tests are typically modelled as normal.

Biology and medicine. Heights, blood pressure, gestation periods, biomarker levels are often approximately normal.

Finance. Returns on a portfolio over a fixed period are sometimes modelled as normal (in basic Methods contexts; real finance uses heavier tails).

Common errors

Standardisation with the wrong denominator. $z = (x - \mu) / \sigma$, not $(x - \mu) / \sigma^2$. Divide by the standard deviation, not the variance.

Forgetting the standardisation when using $Z$-tables. If you look up $z$ directly using $x$, you get the wrong probability.

Inverse direction error. "$P(X > c) = 0.10$" means $c$ is in the upper tail; "$P(X < c) = 0.10$" means $c$ is in the lower tail. Use $1 - p$ if needed before invNorm.

Empirical rule misapplied. The empirical rule is for $\mu \pm n\sigma$ specifically. For other endpoints, standardise and use a table or calculator.

Using $\sigma^2$ where $\sigma$ is asked for. The standard deviation is $\sigma$, not $\sigma^2$. Remember to take the square root if you computed variance.

Calculator without set-up. Paper 2 expects the standardisation or normCdf set-up shown explicitly, with the calculator-derived value at the end. A bare number loses marks.

In one sentence

The normal distribution $N(\mu, \sigma^2)$ is the bell-shaped continuous distribution with mean $\mu$ and standard deviation $\sigma$, and probabilities are computed by standardising via $Z = (X - \mu)/\sigma$ and reading from the standard normal $Z \sim N(0,1)$; the empirical 68/95/99.7 rule covers Paper 1 exact-value questions and the inverse-normal function handles Paper 2 inverse probability questions.