Uni-Sine
New Zealand exams
New Zealand examsNCEA Level 3

NCEA Level 3 Statistics formula sheet (formulae and tables booklet)

Published 6 min read

Paper resets in

Formula sheet

The card below opens today's Level 3 Statistics paper with the sheet in the drawer. The contents follow in our own layout. This is the 2025 booklet, the latest NZQA has published. NZQA's 2026 specification says a Formulae and Tables Booklet will be provided again, so expect the same layout.

Formula sheet

NCEA Level 3 Statistics

Open formula sheet

Permutations and combinations

^nP_r=\frac{n!}{(n-r)!}Permutations
\binom{n}{r}=\ ^nC_r=\frac{n!}{(n-r)!r!}Combinations

Expectation algebra

X and Y are independent for the variance rules

E[aX+b]=aE[X]+bMean of aX + b
Var[aX+b]=a^2Var[X]Variance of aX + b
E[aX+bY]=aE[X]+bE[Y]Mean of aX + bY
Var[aX+bY]=a^2Var[X]+b^2Var[Y]Variance of aX + bY

Probability

P(A\cup B)=P(A)+P(B)-P(A\cap B)Addition rule
P(A|B)=\frac{P(A\cap B)}{P(B)}Conditional probability

Discrete random variables

\mu=E(X)=\Sigma x\cdot P(X=x)Mean
\sigma^2=Var(X)=\Sigma(x-\mu)^2\cdot P(X=x)=E(X^2)-[E(X)]^2Variance
\sigma=SD(X)=\sqrt{Var(X)}Standard deviation

Continuous distributions

f(x)=\frac{1}{b-a}Uniform distribution: density for a ≤ x ≤ b (0 elsewhere)
f(x)=\frac{2(x-a)}{(b-a)(c-a)}Triangular distribution: density for a ≤ x ≤ c
f(x)=\frac{2(b-x)}{(b-a)(b-c)}Triangular distribution: density for c ≤ x ≤ b (0 outside a to b)
A=\frac{1}{2}\times\text{base}\times\text{height}Area of a triangle

Standard normal, Poisson and binomial

Each is printed with a table of probabilities

z=\frac{x-\mu}{\sigma}Standardising
P(X=x)=\frac{\lambda^xe^{-\lambda}}{x!}Poisson probability
\mu=\lambda,\ \sigma=\sqrt{\lambda}Poisson mean and standard deviation
P(X=x)=\binom{n}{x}\pi^x(1-\pi)^{n-x}Binomial probability
\mu=n\pi,\ \sigma=\sqrt{n\pi(1-\pi)}Binomial mean and standard deviation

Printed tables

TableWhat it covers
Standard normal tablez from 0.0 to 3.9 down the side and 0 to 9 across, with a differences block, giving the probability that Z lies between 0 and z
Poisson tableλ from 0.1 to 6.0 across and x from 0 to 17 down, giving P(X = x)
Binomial tablen from 4 to 10, with π across as 0.05, 0.1, 0.15, 1/6, 0.2, 0.25, 0.3, 1/3, 0.35, 0.4, 0.45 and 0.5, giving P(X = x)

What is not on the sheet

  • The margin of error rule of thumb. NZQA's specification says candidates should recall and use the rules of thumb based on 1/√n for margin of error in 91584. They are not printed.
  • The multiplication rule and independence. The sheet has the addition rule and conditional probability. You rearrange conditional probability to get P(A ∩ B) yourself.
  • Negative z values. The normal table starts at 0, so use the symmetry of the curve.
  • Cumulative probabilities. The Poisson and binomial tables give P(X = x). For P(X ≤ x) you add rows.
  • Assumptions. NZQA says to identify the distribution and state its assumptions. The sheet will not tell you that bookings are independent or events occur at a constant rate.
  • Report evaluation. 91584 is about reading statistically based reports. The booklet has nothing for that.

NZQA also says an approved calculator, preferably a graphing calculator, is needed, that questions are set in real-life contexts, and that sensible rounding is expected, with early rounding penalised.

How to use the sheet in the exam

  1. Name the distribution and say why: a count of independent events at a steady rate is Poisson, a fixed number of yes or no trials is binomial.
  2. Write the parameters before you look anything up.
  3. Decide between the formula and the table. A table is faster, but only where its n, π or λ appears.
  4. State your answer in context, with a sensible number of decimal places.

Binomial: formula and table agree

Each of 8 seats on a shuttle is taken independently with probability 0.3. Find the probability that exactly 3 are taken. The sheet gives the binomial line with n = 8, π = 0.3 and x = 3:

P(X=3)=\binom{8}{3}(0.3)^3(0.7)^5=56\times0.027\times0.16807=0.2541

The binomial table gives the same value, 0.2541, in the n = 8 block at x = 3 under π = 0.3. The table saves the arithmetic, but you still need the formula when n is 11 or π is not in the table.

Expectation algebra

A random variable X has a mean of 12 and a variance of 9. Let Y = 3X − 5. The expectation algebra lines give:

E[Y]=3E[X]-5=3\times12-5=31\qquad Var[Y]=3^2Var[X]=9\times9=81\qquad SD(Y)=\sqrt{81}=9

The −5 shifts the mean but does not change the spread, and the 3 multiplies the standard deviation by 3. Practise reading the normal table in our Probability course, then press Mark to check your answer there:

Pick the empirical-rule percentage for a shaded region of a normal curve.

Press Mark to have your answer marked in the Probability course.

Normal: using the table

Journey times are normal with a mean of 70 minutes and a standard deviation of 8 minutes. Find the probability that a journey takes between 70 and 82 minutes. First standardise: z = (82 − 70) ÷ 8 = 1.5. The table entry for 1.5 gives the area between 0 and z, which is 0.4332. A journey over 82 minutes is then 0.5 − 0.4332 = 0.0668.

The Level 3 sheet and the others

The Level 2 Maths sheet has the same standard normal table but no Poisson or binomial, and no probability formulae. For how the Level 3 Statistics standards work, see the NCEA Level 3 Statistics guide.

Exam answers

The answers and a worked mark scheme come with today's paper once it is marked. Sit the paper with the booklet open beside you, mark it, and the worked answers show which line or table each question wanted. We do not publish official or past paper answers. Every question on the daily paper is our own.

Use the sheet with today's paper

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. Open the sheet from the paper itself and practise choosing between formula and table.

Common mistakes with the Level 3 Statistics sheet

  • Choosing the wrong distribution. Binomial has a fixed number of trials. Poisson counts events in an interval.
  • Reading the normal table as the area below z. It gives the area from 0 to z.
  • Forgetting independence. The variance rule for aX + bY needs X and Y independent.
  • Rounding early. NZQA says early rounding may be penalised.

Common questions

Do you get a formula sheet in NCEA Level 3 Statistics?
Yes. NZQA provides a Formulae and Tables Booklet, L3-STATF, with the exam for 91584, 91585 and 91586. You may keep it.
When is the 2026 Level 3 Statistics exam?
Tuesday 24 November 2026, in the morning session, which starts at 9.30 am.
Are the normal, binomial and Poisson tables provided in Level 3 Statistics?
Yes. All three are printed in the booklet, with the formulae for each.
Is the margin of error formula on the sheet?
No. NZQA says candidates should recall the rules of thumb based on 1/√n.

Revision cards

Say the answer first, then flip the card to check it.

Where to go next

  • Solve it

    Probability Calculator

    Visualise probability distributions and events

    Open the tool
  • Learn it

    Probability

    Distributions, expected values and the maths behind uncertainty

    Start the course
  • Test it

    NCEA Level 3 Statistics

    A new exam-style paper every day, with instant marking and a mark scheme

    Start today's paper
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.