NCEA 91586: apply probability distributions in solving problems (Level 3 Statistics)

What NCEA 91586 assesses

NZQA's standard says you apply probability distributions to problems set in real-life or statistical contexts, using methods from:

• discrete and continuous probability distributions
• the mean and standard deviation of random variables
• the distribution of true probabilities versus model estimates versus experimental estimates

NZQA's specification names the distributions: normal, Poisson, binomial, uniform and triangular. You may be asked to calculate or estimate the mean and standard deviation of a random variable, and to identify the distribution you used and state your assumptions. Achieved selects and uses methods. Merit uses relational thinking: a logical sequence of steps, connecting representations, and relating findings to the context. Excellence uses extended abstract thinking: devising a strategy, a chain of reasoning or a generalisation, and using context to reflect on the answer.

The NCEA 91586 exam in 2026 and 2027

NCEA 91586 is one of three Level 3 Statistics externals, with 91584 and 91585. The 2026 exam is on Tuesday 24 November, in the morning session. NZQA's 2027 timetable is not out yet: it says exams run from 9 November to 3 December 2027. The 2026 and 2027 specifications match: printed paper, a Formulae and Tables Booklet, and an approved calculator, preferably a graphing calculator.

• Name the distribution. Say which one you are using and why it fits, for example "binomial: fixed number of independent trials, constant probability".
• State assumptions. The specification asks for them, and Excellence often hinges on judging whether they are reasonable.
• Probabilities can come from anywhere. Formulae, tables, graphs, simulation results or written information.

How NCEA 91586 is marked: grade score marking

Each question scores N0 to E8 and the three add to 24. Choosing and naming the right distribution earns the early units, and judging whether its assumptions hold in context supplies the Merit and Excellence units.

NCEA 91586 cut scores

These are NZQA's cut scores for the total out of 24, taken from each year's assessment schedule. Over these five years Achieved needed 7 or 8, Merit 13 or 14, and Excellence always 19. The 2026 cut scores will appear in NZQA's 2026 schedule once the exam has been marked.

Worked examples at each grade

Achieved: a binomial probability

A player scores with each of 10 independent shots with probability 0.6. Find the probability of exactly 7 scores. The number of scores X is binomial with n = 10 and p = 0.6:

The mean is 10 × 0.6 = 6 and the standard deviation is √(10 × 0.6 × 0.4) = 1.55.

Merit: a Poisson model and what it means

A help line receives calls at an average of 3 per 10 minutes, at random. The number of calls in a 10-minute period is Poisson with mean 3, so the probability of at least one call is:

For 25 minutes the mean scales to 3 × 2.5 = 7.5. Then P(X ≥ 6) = 1 − P(X ≤ 5) = 1 − 0.241 = 0.759. Merit asks you to link the steps and say what this means: on about three nights in four, the line receives six or more calls in 25 minutes. The model assumes calls are independent and the rate is constant, which is unlikely to hold over a whole evening.

Excellence: find the parameters, then judge the model

Journey times are normal with a mean of 40 seconds, and 10% of journeys take longer than 52 seconds. Find the standard deviation. The upper 10% point has z = 1.2816:

Then P(X < 30) = P(Z < −1.07) = 0.143. Working backwards from a percentage to a parameter, and then using it, is the kind of strategy Excellence rewards. Finish by questioning the model: times cannot be negative, so a normal distribution is only a reasonable model if the mean is several standard deviations above zero, which it is here (40 ÷ 9.36 is about 4.3).

Now try a normal distribution question in our Probability course, then press Mark to check your answer in the course:

Exam answers

The answers and a worked mark scheme come with today's paper once it is marked. We do not publish official or past paper answers. Every question on the daily paper is our own.

Use the daily paper as a practice exam

Uni-Sine publishes a free daily paper on the NCEA Level 3 Statistics track. It is a short timed paper of 8 questions worth 25 to 35 marks, not a full NCEA exam, and it is not written by NZQA. Its topics include binomial distribution and normal distribution, which are two of the distributions in 91586. Use Poisson, uniform and triangular questions from your class notes alongside it. For the whole subject, see the NCEA Level 3 Statistics guide.

Common mistakes in NCEA 91586

• No justification. NZQA's 2025 report says a correct number alone is not enough for higher grades: you need calculations and reasoning linked to the context.
• Blank parts. The report says a significant number of candidates leave parts blank.
• Rounding early. The report also flags early rounding and weak calculator skills, including standard form.
• Forgetting the assumptions. Name them, and say whether they are reasonable.

Read next

New Zealand · NCEA Level 3NCEA Level 3 Statistics: standards, exam date and practiceNCEA Level 3 Statistics explained: the internals, the three externals on reports, probability and distributions, the 2026 exam date and worked examples.