Scottish examsAdvanced Higher

Advanced Higher Statistics formula sheet: Statistical Formulae and Tables

Published 12 Nov 20266 min read

Higher Applications of Maths

Paper resets in

Formula sheet

The card below opens today's paper with this booklet in the drawer. The contents follow in our own layout. Check against the official booklet, which also holds the numerical values in each of the 8 tables, before you rely on this page.

Formula sheet

Higher Applications of Maths

Open formula sheet

Probability distributions

P(X=x)=\frac{1}{k}discrete uniform probability function

P(X=x)={}^nC_xp^x(1-p)^{n-x}binomial probability function

P(X=x)=\frac{\lambda^xe^{-\lambda}}{x!}Poisson probability function

f(x)=\frac{1}{b-a}continuous uniform density

f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}normal density

Distribution	Notation	Mean	Variance
Discrete uniform	U(k)	(k + 1) ÷ 2	(k² − 1) ÷ 12
Binomial	B(n, p)	np	np(1 − p)
Poisson	Po(λ)	λ	λ
Continuous uniform	U(a, b)	(a + b) ÷ 2	(b − a)² ÷ 12
Normal	N(μ, σ²)	μ	σ²

Control charts: the Western Electric Company rules

A process may be out of statistical control if any of these happen:

Rule	Signal
1	Any single point falls outside a 3σ limit
2	Two out of three consecutive points fall beyond the same 2σ limit
3	Four out of five consecutive points fall beyond the same 1σ limit
4	Eight consecutive points fall on the same side of the centre line

Sums of squares and products

S_{xx}=\sum(x_i-\overline{x})^2=\sum x_i^2-\frac{(\sum x_i)^2}{n}sum of squares of x

S_{yy}=\sum(y_i-\overline{y})^2=\sum y_i^2-\frac{(\sum y_i)^2}{n}sum of squares of y

S_{xy}=\sum(x_i-\overline{x})(y_i-\overline{y})=\sum x_iy_i-\frac{\sum x_i\sum y_i}{n}sum of products

s=\sqrt{\frac{S_{xx}}{n-1}}sample standard deviation

Correlation

r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}product moment correlation coefficient

Linear regression

The linear model has independent errors with mean 0 and variance σ².

Y_i=\alpha+\beta x_i+\varepsilon_ithe linear model

b=\frac{S_{xy}}{S_{xx}}least squares estimate of the slope

a=\overline{y}-b\overline{x}least squares estimate of the intercept

SSR=S_{yy}-\frac{S_{xy}^2}{S_{xx}}sum of squared residuals

s^2=\frac{SSR}{n-2}estimate of the error variance

\hat{Y}_i\pm t_{n-2,1-\alpha/2}\ s\sqrt{1+\frac{1}{n}+\frac{(x_i-\overline{x})^2}{S_{xx}}}prediction interval for an individual value of Y at x

\hat{Y}_i\pm t_{n-2,1-\alpha/2}\ s\sqrt{\frac{1}{n}+\frac{(x_i-\overline{x})^2}{S_{xx}}}confidence interval for the mean of Y at x

Hypothesis test statistics

z=\frac{\overline{X}_1-\overline{X}_2-(\mu_1-\mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}z-test for a difference in means

t=\frac{\overline{X}_1-\overline{X}_2-(\mu_1-\mu_2)}{s\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}t-test for a difference in means

s^2=\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}pooled variance, with n₁ + n₂ − 2 degrees of freedom

z=\frac{\hat{p}_1-\hat{p}_2}{\sqrt{\hat{p}\hat{q}\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}}z-test for a difference in proportions

\hat{p}=\frac{n_1\hat{p}_1+n_2\hat{p}_2}{n_1+n_2}the pooled proportion

\sum\frac{(O_i-E_i)^2}{E_i}chi-squared test statistic, for goodness of fit and contingency tables

t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}test of zero correlation

t=\frac{b\sqrt{S_{xx}}}{s}test of zero slope

Statistical tables

Table	What it gives
1	Binomial cumulative distribution function
2	Poisson cumulative distribution function
3	Standard normal cumulative distribution function
4	Percentage points of the standard normal distribution
5	Student t distribution
6	Chi-squared distribution
7	Wilcoxon signed-rank test
8	Mann-Whitney test

What is not in the Advanced Higher Statistics booklet

Which test to use. The booklet gives test statistics, but never says whether a question needs a z-test, a t-test, a chi-squared test or a Wilcoxon test. Naming the right one is a large part of the marks.
Hypotheses and conclusions. You write H₀ and H₁, state the significance level, compare with the critical value and conclude in the context of the question.
Assumptions. Normality, independence and equal variances are for you to state and check.
Probability basics. The addition and multiplication laws, conditional probability and Bayes' theorem are not printed.
Summary statistics. The mean, the median, quartiles and the method for outliers are not given.
Confidence intervals for a mean or proportion. The booklet prints the regression intervals, not the ordinary ones, so learn those.
Experimental design. Blocking, randomisation and sampling methods are for you to explain.

The course specification lists the skills the papers sample, and it points to the booklet for the t-distribution table.

Practise choosing the test in the Hypothesis Test Calculator and the normal distribution questions in the Probability course, then press Mark.

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

How the papers use the booklet

Advanced Higher Statistics is two question papers and no project. The course specification gives 30 marks and 1 hour for Paper 1 and 90 marks and 2 hours 45 minutes for Paper 2, which is 120 marks. A calculator is allowed in both. In the 2027 timetable both are on Wednesday 2 June: Paper 1 from 09:00 to 10:00 and Paper 2 from 13:00 to 15:45 (Exam Timetable 2027, version 02).

Paper	Marks	Time	Calculator
Paper 1	30	1 hour	Yes
Paper 2	90	2 hours 45 minutes	Yes

Our Advanced Higher Statistics guide covers the course and the grades.

Using the formula sheet under exam pressure

Poisson probability

Cars pass a checkpoint at an average of 3 a minute, modelled by a Poisson distribution with λ = 3. Find the probability that exactly 2 pass in a minute.

P(X=2)=\frac{3^2e^{-3}}{2!}=\frac{9\times0.04979}{2}=0.224

Mean and variance of a binomial

If X is B(20, 0.3), the booklet gives mean np and variance np(1 − p):

E(X)=20\times0.3=6,\qquad V(X)=20\times0.3\times0.7=4.2

Regression from the sums of squares

For n = 8 pairs, suppose S_xx = 36, S_xy = 45 and S_yy = 81, with x̄ = 10 and ȳ = 20. The slope and intercept are b = 45 ÷ 36 = 1.25 and a = 20 − 1.25 × 10 = 7.5, so the line is y = 7.5 + 1.25x. The correlation is

r=\frac{45}{\sqrt{36\times81}}=\frac{45}{54}=0.833

For the sum of squared residuals,

SSR=81-\frac{45^2}{36}=24.75,\qquad s^2=\frac{24.75}{8-2}=4.125,\qquad s=2.03

Exam answers

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

Use the sheet with today's paper

Uni-Sine publishes a free daily paper for Higher Applications of Maths, which is the closest statistics track. It is a short timed paper of 8 questions worth 25 to 35 marks, not a full SQA paper and not set by SQA. Many schools set a longer prelim in winter: see our prelims guide. Open the sheet, set a timer, and write the line you use at the start of each answer.

Common mistakes

Using the wrong standard deviation. The booklet divides by n − 1 for the sample standard deviation.
Mixing up the two intervals. The prediction interval for an individual value has an extra 1 under the root, so it is wider than the interval for the mean.
Using n instead of n − 2. Regression uses n − 2 degrees of freedom.
Ignoring the conditions for chi-squared. At least 80% of the expected values should be at least 5, and none below 1.
Stopping at the number. A test result needs a conclusion in the words of the question.

Common questions

Do you get a formula sheet in Advanced Higher Statistics?

Yes. SQA publishes a booklet called Statistical Formulae and Tables for the course. It holds the formulae and 8 statistical tables.

What is in the Advanced Higher Statistics formulae and tables?

Distribution formulae, the Western Electric Company rules, sums of squares, correlation, regression, hypothesis test statistics, and tables for the binomial, Poisson, normal, t, chi-squared, Wilcoxon and Mann-Whitney.

Is a calculator allowed in Advanced Higher Statistics?

Yes, in both papers.

Does the booklet say which test to use?

No. It gives the statistics, and you decide which test the question needs.

When is the Advanced Higher Statistics exam in 2027?

Wednesday 2 June 2027: Paper 1 from 09:00 and Paper 2 from 13:00.

Revision cards

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

Where to go next

Solve it
Hypothesis Test Calculator
Run z, t and chi-squared tests and read off the p-value
Open the tool
Learn it
Probability
Distributions, expected values and the maths behind uncertainty
Start the course
Test it
Higher Applications of Maths
A new exam-style paper every day, with instant marking and a mark scheme
Start today's paper