The normal distribution: the two calculator moves that unlock every question

Every S1 paper has 6-10 marks of normal distribution. Every one of those marks is accessible if you know two calculator moves cold. Here they are, on one page.

Move 1 — Finding a probability

Question shape: "$X \sim N(\mu, \sigma^2)$. Find $P(a < X < b)$."

Casio CG50 keystrokes:

• Menu → STAT
• DIST → NORM → Ncd (normal cumulative distribution)
• Lower: $a$
• Upper: $b$
• $\sigma$: standard deviation (NOT variance)
• $\mu$: mean

Press EXE. The probability is on-screen.

Common trap: entering $\sigma^2$ instead of $\sigma$. The question gives you the variance; the calculator wants the standard deviation. Take the square root FIRST.

For $P(X < a)$: set Lower to a very negative number (e.g. $-10^{99}$). The keystroke on Casio is (-)(EXP)99.

For $P(X > a)$: set Upper to a very large number (e.g. $10^{99}$). Keystroke: EXP 99.

Move 2 — Finding a value from a probability

Question shape: "$X \sim N(\mu, \sigma^2)$. Find $a$ such that $P(X < a) = 0.85$."

This is the INVERSE normal — one of the least-known and highest-yielding calculator moves in the S1 syllabus.

Casio CG50 keystrokes:

• Menu → STAT
• DIST → NORM → InvN (inverse normal)
• Area: 0.85 (the given probability)
• $\sigma$: standard deviation
• $\mu$: mean
• Tail: LEFT (for "$P(X < a)$"), RIGHT (for "$P(X > a)$"), or CNTR (for two-tailed).

Press EXE. The value of $a$ is on-screen.

The full workflow

Every normal-distribution question fits one of these two moves. Read the question:

• Does it give you a value ($a$) and ask for a probability? → Move 1 (Ncd).
• Does it give you a probability and ask for a value ($a$)? → Move 2 (InvN).
• Does it give you a mean and standard deviation and ask both? Do them separately.

The three-mark example

*"$X \sim N(50, 100)$. Find $P(X > 65)$."*

• $\sigma^2 = 100$ → $\sigma = 10$.
• Ncd, Lower = 65, Upper = $10^{99}$, $\sigma = 10$, $\mu = 50$.
• Answer: $P(X > 65) \approx 0.0668$.

Total time: 30 seconds.

The four-mark inverse example

*"$X \sim N(80, \sigma^2)$. $P(X > 90) = 0.10$. Find $\sigma$."*

• This is inverse — we know the probability (0.10) and need to work backwards.
• Standardise: $Z = \dfrac{90 - 80}{\sigma}$; $P(Z > z) = 0.10 \Rightarrow z \approx 1.2816$ (InvN, Area 0.9, Tail Left, $\sigma = 1$, $\mu = 0$).
• So $\dfrac{10}{\sigma} = 1.2816 \Rightarrow \sigma \approx 7.80$.

Yes — you use InvN on the STANDARD normal (mean 0, sd 1) first to find $z$, then solve for $\sigma$. This is the third-most-common exam pattern and it catches half the cohort every year.

Setting up your calculator

Two settings to check before the paper:

1. Angle: RADIAN (for calculus questions) or DEGREE (for stats questions with trig). Toggle via SETUP.

2. Fraction format: display fractions (d/c) or decimals (a b/c) — your choice, but stick to one throughout the paper for consistency.

Your drill

Print the two moves. Do six normal-distribution questions from past papers using ONLY these two moves. Time yourself — target: under 3 minutes per question. If you're going over, you're second-guessing the input. Trust the calculator.

Two moves, six questions, 45 minutes of drill. Ten marks unlocked forever.

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