In IB Maths AI SL a GDC is allowed on both papers, so normal distribution questions are GDC questions: you are never asked to use tables. That makes the calculation the easy part. The marks that go missing are for communication: which probability you found, which distribution you used, and whether the answer is sensible. This post covers the three calculations you need, the buttons on the TI-84 Plus CE, TI-Nspire and Casio fx-CG50, and what to write down.
The three calculations
Every AI SL normal distribution question is one of these, or a combination of them:
| You are given | You want | GDC function |
|---|---|---|
| a value of $X$ | a probability | normal cumulative distribution (normalcdf / Ncd) |
| a probability | a value of $X$ | inverse normal (invNorm / InvN) |
| a probability and a sample size | an expected number | normal cdf, then multiply by $n$ |
Throughout, $X \sim N(\mu, \sigma^2)$: the second number is the variance. Your GDC wants the standard deviation $\sigma$. If a question says $X \sim N(165, 64)$, type $\sigma = 8$, not $64$.
The buttons
TI-84 Plus CE. Press 2nd then VARS (DISTR). normalcdf(lower, upper, μ, σ) gives a probability. invNorm(area, μ, σ, tail) gives a value of $X$; set the tail to LEFT, CENTER or RIGHT to match the area you typed.
TI-Nspire. In a Calculator page: menu, Probability, Distributions, then Normal Cdf (Lower Bound, Upper Bound, μ, σ) or Inverse Normal (Area, μ, σ), where the area is to the left of the value.
Casio fx-CG50. MENU, Statistics, DIST (F5), NORM (F1), then Ncd for a probability (Lower, Upper, $\sigma$, $\mu$) or InvN for a value (Tail Left, Right or Central, Area, $\sigma$, $\mu$). The Casio asks for $\sigma$ before $\mu$, the opposite order to the TI calculators, so read each prompt.
For "less than" use a lower bound of $-10^{99}$ and for "greater than" an upper bound of $10^{99}$ (entered as 1E99 with the EE or EXP key).
Worked examples
The heights of students at a school are modelled by $H \sim N(165, 8^2)$, in cm.
Example 1 · a lower-tail probability · 2 marks
Find the probability that a randomly chosen student is shorter than $150$ cm.
Solution
Write what you are finding: $P(H<150)$.
GDC: normal cdf with lower $-10^{99}$, upper $150$, $\mu=165$, $\sigma=8$.
$P(H<150) = 0.0304$ (3 s.f.)
Marks: (M1) for the correct probability statement or a sketch with the region shaded, (A1) for $0.0304$.
Example 2 · a probability between two values · 2 marks
Find the probability that a randomly chosen student is between $155$ cm and $175$ cm tall.
Solution
$P(155<H<175)$: normal cdf with lower $155$, upper $175$, $\mu=165$, $\sigma=8$.
$P(155<H<175) = 0.789$ (3 s.f.)
Sense check: $155$ and $175$ are both $1.25$ standard deviations from the mean, so the answer should be a large, central probability. It is.
Example 3 · an expected number · 3 marks
There are $400$ students at the school. Find the expected number of students taller than $180$ cm.
Solution
$P(H>180)$: normal cdf with lower $180$, upper $10^{99}$, $\mu=165$, $\sigma=8$, gives $0.030396\ldots$
Expected number $= 400 \times 0.030396\ldots = 12.2$ (3 s.f.)
Two points about Example 3. First, $P(H>180)$ equals $P(H<150)$ from Example 1, because $150$ and $180$ are both $15$ cm from the mean: symmetry is a quick check on your GDC entry. Second, an expected number does not have to be a whole number. Give $12.2$ unless the question asks you to round.
Example 4 · inverse normal · 3 marks
The tallest $5\%$ of students are invited to try out for the basketball team. Find the minimum height needed to be invited.
Solution
The tallest $5\%$ is the right tail, so the area to the left of the height $h$ is $1-0.05=0.95$.
$P(H<h)=0.95$: inverse normal with area $0.95$, $\mu=165$, $\sigma=8$.
$h = 178$ cm (3 s.f.)
The tail trap
Typing $0.05$ into inverse normal finds the height that the shortest $5\%$ are below, about $152$ cm. If your answer is on the wrong side of the mean, you used the wrong tail. Either convert to a left area ($1-0.05$) or, if your calculator has a tail option, set it to Right.
What to write so the method marks are yours
A bare number from the GDC can earn full marks when it is correct. When it is wrong, it earns nothing, because the examiner cannot see a method to credit. On every normal distribution question, write:
- The probability statement, such as $P(H<150)$ or $P(H<h)=0.95$. This often carries a method mark on its own.
- A sketch of the bell curve with the region shaded, for anything involving two bounds or an inverse.
- The answer to 3 significant figures, unless the question says otherwise, keeping the unrounded value for later parts.
Avoid writing calculator syntax such as normalcdf(-1E99,150,165,8) as your only working. Mathematical notation is what the mark scheme looks for.
Where this appears on the paper
Normal distribution parts often sit inside a longer question with a binomial part: "find the probability that a student is taller than 180 cm; ten students are chosen at random; find the probability that exactly two are taller than 180 cm". The first answer becomes $p$ in $X \sim B(10, p)$. Use the unrounded value of $p$ for the second part.
You can practise both distributions, easy to hard, on the AI SL probability distributions questions page, and the AI SL formulas page collects the course formulas in one place; check them against your own copy of the official formula booklet.
Practise this topic
FAQ
Do I need z-scores for IB Maths AI SL?
For probabilities and inverse normal questions with a known mean and standard deviation, no. Your GDC does the calculation directly. You do need to write down what you are calculating, for example $P(H<150)$.
What do I type for infinity on the GDC?
A very large number, such as $10^{99}$ (1E99). For a lower tail use $-10^{99}$ as the lower bound. Any bound far beyond the data gives the same answer to 3 significant figures.
How many decimal places should normal distribution answers have?
Give answers exactly or to 3 significant figures unless the question says otherwise. Keep the full GDC value if you use it in a later part.
Exam-style AI SL normal and binomial distribution questions, easy to hard, each with a step-by-step mark scheme.
Practise AI SL probability distributions →