QCE Mathematical Methods - Unit 4 - Continuous random variables and the normal distribution

Normal Distributions | QCE Mathematical Methods

Learn QCE Mathematical Methods normal distributions, z-scores, probabilities, quantiles and standardisation.

Updated 2026-05-18 - 4 min read

The normal distribution is the familiar bell-shaped distribution used for many naturally varying measurements: heights, errors, measurement noise and standardised scores. It is not appropriate for every dataset, but it is one of the most important continuous models in Methods.

Notation

If $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$, write:

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

The mean $\mu$ locates the centre. The standard deviation $\sigma$ controls the spread. A larger $\sigma$ makes the bell curve wider and lower because the total area must still be $1$.

The probability density function is:

$ f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} $

You do not usually integrate this by hand in Methods, but the formula explains the key graph features: the curve is symmetric about $\mu$, always positive, bell-shaped and controlled by $\sigma$.

The standard normal distribution is:

$ Z\sim N(0,1) $

It is the reference normal distribution after standardising.

Standardising with z-scores

A $z$-score tells you how many standard deviations a value is from the mean:

$ z=\frac{x-\mu}{\sigma} $

If $z=2$, the value is two standard deviations above the mean. If $z=-1.5$, the value is one and a half standard deviations below the mean.

Standardising lets you compare values from different normal distributions.

You can also reverse the formula:

$ x=\mu+z\sigma $

This is useful for percentile and quantile questions. Once technology gives a $z$-value, convert it back to the original scale.

Reading the graph

Changing $\mu$ shifts the curve left or right. Changing $\sigma$ changes the spread:

| Change | Effect on graph | What stays true | | --- | --- | --- | | larger $\mu$ | centre moves right | total area is still $1$ | | smaller $\mu$ | centre moves left | shape stays symmetric | | larger $\sigma$ | curve becomes wider and lower | centre stays at $\mu$ | | smaller $\sigma$ | curve becomes narrower and taller | total area is still $1$ |

If the data are strongly skewed, bounded in a way the normal model cannot respect, or contain several clusters, a normal model may be a poor choice even if a calculator can still produce a number.

Probabilities and quantiles

For normal distributions, QCE questions usually expect technology for probabilities and quantiles. Distribution tables are not required in the current syllabus.

• Probability question: find an area, such as $P(X<75)$.
• Quantile question: find the value of $x$ that cuts off a given area, such as the 90th percentile.

For a middle interval:

$ P(a<X<b)=P(X<b)-P(X<a) $

For a right-tail probability:

$ P(X>a)=1-P(X\le a) $

For a quantile, start with the area. If a question asks for the top $10\%$, the cutoff has left-tail area $0.90$, not $0.10$.

Worked example

Worked probability example

Worked quantile example

Comparing scores

Suppose one student scores $86$ on a test with mean $80$ and standard deviation $4$. Another scores $72$ on a test with mean $60$ and standard deviation $10$.

First student:

$ z=\frac{86-80}{4}=1.5 $

Second student:

$ z=\frac{72-60}{10}=1.2 $

Relative to their cohorts, the first score is stronger because it is further above the mean in standard-deviation units.

Common mistake

Quick check

Sources

• QCAA Mathematical Methods subject page
• QCAA Mathematical Methods 2025 syllabus
• OpenStax Introductory Statistics 2e: The Standard Normal Distribution
• OpenStax Introductory Statistics 2e: Using the Normal Distribution