Year 12 / Continuous random variables and normal distributions

Standardise the distance.

A normal model is symmetric about its mean μ, with spread measured by standard deviation σ. In X ∼ N(μ, σ²), the second parameter is the variance. Standardising with z = (x − μ)/σ converts to a standard normal variable.

Calculate interval probabilities as differences of cumulative probabilities. To find a percentile, use the inverse cumulative distribution then transform back with x = μ + zσ. The normal model is an assumption to assess against the context and the data.

INTERACTIVE MODEL

Try it. Watch it change.

N(0,

1.3^{2}

)

Mean μ0Standard deviation σ1.3Upper boundary x1

z = 0.7692 · P(X ≤ x) ≈ 0.7791

The probability includes the full left tail. The plotted window is finite; probabilities use the normal CDF, not a cropped area estimate.

Explore: Change the mean and spread, then move the probability boundary. Compare the shaded area with the z-score.

Standard score

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

Transform back

x=\mu+z\sigma

WORKED EXAMPLE

X ∼ N(100, $15^{2}$ ). Standardise x = 130.

z = (130 − 100)/15.
z = 2, so this value is two standard deviations above the mean.

Assumed knowledge

Density, mean, variance and calculator normal functions.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Use normal-model parameters and standard scores.
Calculate probabilities and quantiles with technology.

QCAA Mathematical Methods 2025 v1.3 · p. 29 ↗

READY TO TRY IT?

Standardise the distance.

X ∼ N(100, 15215^{2}152). Standardise x = 130.

Put the idea to work.

X ∼ N(100, $15^{2}$ ). Standardise x = 130.