Examinations Syllabus for 2015

Contents

Page
Summary of changes from previous version of syllabus .......................................................... 2
Introduction .......................................................................................................................... 3
Ordinary Certificate in Statistics ............................................................................................ 4
Module 1 ............................................................................................................................ 5
Module 2 ............................................................................................................................ 6
Higher Certificate in Statistics ............................................................................................... 8
Module 1 .......................................................................................................................... 10
Module 2 .......................................................................................................................... 12
Module 3 .......................................................................................................................... 13
Module 4 .......................................................................................................................... 14
Module 5 .......................................................................................................................... 15
Module 6 .......................................................................................................................... 16
Module 7 .......................................................................................................................... 17
Module 8 .......................................................................................................................... 18
Advised academic progression through the modules ......................................................... 19
Mathematics background.................................................................................................. 20
Interpretation of computer output ..................................................................................... 20
Table of mathematics requirements for each module ......................................................... 21
Graduate Diploma in Statistics ............................................................................................ 22
Module 1 .......................................................................................................................... 24
Module 2 .......................................................................................................................... 26
Module 3 .......................................................................................................................... 28
Module 4 .......................................................................................................................... 29
Module 5 .......................................................................................................................... 31
Mathematics Background ................................................................................................. 33

1

Summary of Changes from Previous Version of Syllabus

Changes to 2015 syllabus compared with 2014 syllabus
Some minor wording changes were made for Higher Certificate Module 6 (Further Applications of
Statistics).

2

Introduction to the Examinations
The Royal Statistical Society’s professional examinations provide a route into the statisticalprofession at introductory, intermediate and advanced levels. They will be suitable for many whowish to enter into or advance within the profession. They may be particularly helpful for those
who wish to convert to statistics from another discipline, and for those in countries where little or
no statistical training and few formal statistical qualifications are available.
The examinations consist of papers at three levels, namely Ordinary Certificate, Higher Certificate
and Graduate Diploma. Taken in sequence, these levels provide a comprehensive syllabus ofstudy in applied statistics.
The Ordinary Certificate is offered in a modular form consisting of two modules which may betaken singly or together. Each module is separately certificated. The Ordinary Certificate as awhole is awarded on successful completion of both modules; the "shelf life" of modules leading
to this Certificate is unlimited.
The Higher Certificate is offered in a modular form consisting of eight modules which may betaken singly or in any combination. Each module is separately certificated. The Higher Certificate
as a whole is awarded on successful completion of six modules (subject to some restrictions on
choice of modules); the "shelf life" of modules leading to this Certificate is unlimited.
The Graduate Diploma is offered in a modular form consisting of five modules which may betaken singly or in any combination. Each module is separately certificated. The Graduate
Diploma as a whole is awarded on successful completion of all modules; the "shelf life" ofmodules leading to the Diploma is unlimited.
The Graduate Diploma is widely recognised and respected, nationally and internationally, by
employers in the public and private sectors. It is also widely recognised by universities assufficient for admission to an MSc course in Statistics.
The examinations are held in May each year. Full details of how to register for the examinations
can be found in the General Guide to Examinations.
The Society provides statistical tables in each examination. Copies of these tables are available
from the RSS website so that candidates may familiarise themselves with the layout.
Candidates are expected to make appropriate use of calculators in the examinations. Calculator
regulations are included in the General Guide to Examinations.
Copies of past papers and solutions and reading lists for the examinations are available from the
RSS website. Additional specimen papers and solutions for the Higher Certificate and Graduate
Diploma are also available.

3

Ordinary Certificate in Statistics
The Ordinary Certificate of the Royal Statistical Society is offered in a modular form. This gives
the opportunity for candidates to proceed at their own pace. The aim of this first level certificate
is to provide a sound foundation knowledge of the principles and practice of statistics withemphasis on practical data collection, presentation and interpretation. It is intended both as an
end in itself in terms of being a first qualification in statistics, and as a basis for further work instatistics, as for example in the more advanced levels of the Society’s examinations. The
individual modules are also valuable free-standing elements in their own right; they may forexample be useful components of a formal or informal continuing professional development
programme. Candidates are assumed to hold a good General Certificate of Secondary Education(GCSE) in mathematics or an equivalent qualification, or relevant work experience.
The two modules cover the essential ideas of statistics in practice. Anyone familiar with themshould be able to carry out supervised statistical work of a routine kind, or be able to apply
statistical methods, at an elementary level, within work of a more general kind. Module 1
includes the key topics of data collection in the field: what data should be collected, who from,and how will the data be captured? It also includes work on data processing, including a brief
introduction to statistical computing. Module 2 encompasses the basic techniques of descriptive
statistics. Two ideas are of paramount importance: what is the most appropriate way to analyse
the data, and how can the message within the data most clearly be communicated?
Candidates may enter for either or both modules at any examination session. Candidates may
resit any module any number of times; the highest mark achieved will stand. The pass mark foreach individual module is 50%. Module marks are "banked" without time limitation.
Each module is examined by a 3-hour written paper containing from 7 to 9 (inclusive) questionsof different lengths. There are no restrictions on the number that may be answered. The marks
available for each question are printed on the examination papers. Candidates will be advised of
their marks for each module taken. Candidates will receive a separate certificate for each
module in which they are successful.
Candidates who are successful in both modules (not necessarily in the same session) will beawarded the Ordinary Certificate in Statistics. The Ordinary Certificate will only be awarded on
the first occasion when a candidate becomes eligible for it. If a candidate, having been awarded
the Ordinary Certificate, takes either or both modules again, advice of the marks earned will be
issued and, if appropriate, a certificate to indicate success in the individual module(s); but theOrdinary Certificate itself will not be re-awarded.

Module Syllabuses
Syllabuses for the two modules are set out on pages 5 to 7.

4

MODULE 1: Collection and Compilation of Data
The origin, use and interpretation of published or Overview of official statistics in a country of theadministrative data.
candidate's choice. What statistical series areproduced, how are the data collected and to what
uses are the data put?

Elementary ideas of sampling methods.
Definitions of population and sampling frame.
Methods of selecting samples (including practicalproblems) and implications of sample size:
simple random sampling, systematic sampling,cluster sampling, quota sampling, stratified
random sampling and multi-stage sampling.

No formulae are required.
Advantages and disadvantages of each method,
including considerations of cost, precision andaccuracy. Use of random number tables.

Pilot surveys, censuses, sample surveys, personal No formulae are required.
interviews, self-completion questionnaires, postal
and telephone enquiries.
Serial surveys longitudinal or cross-sectional. Problems arisingin the collection of data, dropouts, late returns,'freak' values and their treatment, ‘cleaning’ data.
Non-sampling errors. Identification and inter- No formulae are required. Candidates may bepretation of bias error (e.g. from non-response, asked to provide examples of how different types
errors in defining the population, enumerator of bias error may arise.
distortion, etc).
Design of simple questionnaires and forms for Candidates should be able to produce their own
collection of data. Formulation, classification and simple questionnaire and data form.
coding of questions, including verification.
Making questionnaires suitable for dataprocessing and analysis; use of missing value
codes.
Distinction
betweenexperimental studies.

observationaland

Use of computers for data storage and retrieval.

Access to a computer when preparing for theexaminations is not necessary.

5

MODULE 2: Analysis and Presentation of Data
Use of rough checks for order of magnitude and Candidates are encouraged to check that theirleading digits in results.
answer 'looks about right'.
Approximation, limits of accuracy, rounding and Error in the result of a simple expression when theaccuracy of recording. Percentages, ratios, rates consistent parts are all rounded.
and linear interpolation. Distinction between
discrete and continuous data.
Construction and uses of frequency tables for one Credit will be given for neatness and clarity ofor more variables and contingency tables. Tables presentation.
for presenting collections of results together withsummary tables of frequencies, relevant averages,
standard deviations, etc.
Graphs and diagrams, their use in analysis and Credit will be given for neatness and clarity ofpresentation. Construction, uses and limitations presentation.
of scatter diagrams, time charts, stem and leaf
diagrams, histograms, bar charts, pie charts,
frequency and cumulative frequency curves andboxplots (box and whisker plots).
Sample measures of location and dispersion.
Arithmetic mean, median, mode, percentiles,
range, inter-quartile range, variance, standard
deviation, coefficient of variation; their uses andlimitations as measures; their calculation fromfrequency tables and raw data;graphical
methods of estimation. Distinction between interand intra-subject variation.

Formulae should be known, except that thedefinition of 'hinges' is not required.

Candidates should be able to estimate percentiles
from a cumulative frequency curve (ogive).

Identification of outliers in a data set, and Outliers should be considered in the context ofappreciation of steps that may be taken to deal the data-set as a whole. Candidates need onlywith them.
use informal criteria for classifying observations as

outliers (as in the construction of boxplots), butshould be able to discuss how they might be dealt
with.

Probability as a measure of uncertainty. Link The long-run concept of probability, e.g. frombetween probability and relative frequency. tossing a coin repeatedly, should be understood.
Allocation of probabilities in 'equally likely' cases.
Mutually exclusive events. Independent events. Permutations and combinations are not required.
Addition and multiplication of probabilities withsimple applications. Use of Venn diagrams andtree diagrams.
Calculation of least squares regression line and its
interpretation. Correlation as a measure of linear
association between two variables. Productmoment correlation coefficient.
Spearman's rank correlation coefficient.

Derivation of the least squares estimates (bycalculus) is not required.
Candidates should know when it is appropriate touse each method.
Consideration of tied ranks will not be expected.

Simple moving averages for detecting trends and Candidates should know when it is appropriate tofor smoothing time series. Seasonal data.
use additive or multiplicative models forseasonality, and how each is calculated.

6

Knowledge of weighted forms of moving average.

No calculation is required.

Simple and weighted averages of price relatives. Knowledge of how and why index numbers are
Construction of aggregate (Paasche, Laspeyres used in real life is assumed.
and Fisher) averages. Simple chain-based indices.
Limitation and use of index numbers e.g. inassessment of productivity and prices.
Interpretation. Translation of written statements Candidates may be asked to explain what the
into tabular forms; simple fallacies, typical data presentation tells the reader.
misleading distortion in popular published graphs.
Answers to questions about tables and charts.
Writing of clear and concise reports on numerical Candidates may be tested on their spelling anddata in different contexts.
grammar as well as their logic.

7

Higher Certificate in Statistics
The Higher Certificate of the Royal Statistical Society is offered in a modular form with eight
modules. This gives the opportunity for candidates to proceed at their own pace. The Certificate
is both as an end in itself in terms of a qualification in statistics at a more advanced level thanthat of the Ordinary Certificate, and as a basis for further work in statistics. In addition, theindividual modules are valuable free-standing elements in their own right; they may for example
be useful components of a formal or informal continuing professional development programme.
Candidates may enter for any number of modules, either singly or in combination, in any
examination session. Candidates may resit any module any number of times; the highest markachieved will stand. The pass mark for each individual module is 50%. Module marks are"banked" without time limitation.
Each module is examined by a 1½-hour written paper containing four questions of which
candidates are asked to answer three. Candidates will be advised of their marks for each moduletaken. Candidates will receive a separate certificate for each module in which they aresuccessful.
Candidates who are successful in six modules including modules 1 to 4 (not necessarily all inthe same session) will be awarded the Higher Certificate in Statistics. The Higher Certificate will
only be awarded on the first occasion when a candidate becomes eligible for it. If a candidate,having been awarded the Higher Certificate, takes any module(s) again or takes any further
module(s) not already taken, advice of the marks earned will be issued and, if appropriate, acertificate to indicate success in the individual module(s); but the Higher Certificate itself will not
be re-awarded.
Candidates who wish to proceed to the Graduate Diploma in Statistics are advised that ideally,
they will have passed Higher Certificate Modules 1 to 6 (not necessarily all in the same session).
Candidates are advised that entry to any of the modules assumes knowledge of the material in
the Ordinary Certificate in Statistics, or equivalent. Candidates should ensure that they arefamiliar with this material before sitting for any modules. This is particularly important forcandidates who wish to attempt to achieve the Higher Certificate in Statistics by taking six
modules (including modules 1 to 4) in the same session.

8

MODULE SYLLABUSES
Syllabuses for the eight modules are set out on pages 10 to 19.
The modules are
Module 1

Data collection and interpretation

Module 2

Probability models

Module 3

Basic statistical methods

Module 4

Linear models

Module 5

Further probability and inference

Module 6

Further applications of statistics

Module 7

Time series and index numbers

Module 8

Survey sampling and estimation

Information on advised academic progression through the modules, on the required mathematics
background and on interpretation of computer output can be found on pages 20-22.

9MODULE 1: Data collection and interpretation
Summarising and interpreting data
Frequency distributions. Numerical and graphicalformsofpresentation
andstatisticalinterpretation.
Scatter diagrams, time charts, stem and leaf
diagrams, histograms, bar charts, pie charts,
frequency and cumulative frequency curves,
boxplots (box and whisker plots), dotplots.
Summary statistics for measures of location,
variability and skewness.

Credit will be given for graphs that are wellpresented, including neat appearance andinclusion of appropriate title and labels.

Sample mean, median, mode, quartiles, semiinter-quartile range, standard deviation, variance,
range.

SurveysTarget and study populations. Sampling frames. Principles and practice, with examples fromcandidates' knowledge and experience.
Problems arising in the collection of data.
Censuses, sample surveys and routine collectionof data at intervals of time.
Design of questionnaires and forms for collecting
data.
Personal and telephone interviews, postal Advantages and disadvantages of these methods.
enquiries, pilot enquiries.
Problems of non-response, bias amonginterviewers, question bias, non-sampling errors.
Simple random sampling. Uses and limitations. Candidates will be expected to know and be able
Estimators for means, totals and proportions and to use, but not to derive, formulae for estimators
the variances of these estimators.
and variances, along with an informal

understanding of standard errors. Use of thefinite population correction is not required.

Use of other practical methods of sampling: No formulae are required.
systematic sampling, cluster sampling, quota
sampling, stratified random sampling and multistage sampling.
Exploratory analysis
Candidates should be prepared to examine a setof data, to choose and carry out suitable methodsof analysis, to answer questions from nonstatistical users and to present the analysis andconclusions in the form of a short report. The
techniques required may be of the simplest kind,e.g. plotting, grouping, transforming or
calculating from the data. Candidates will be
expected to use box-plots and other similargraphical displays.

Particular questions may be specified, and also
particular types of non-statistical user (seebelow). Otherwise a brief general report on themain points in the data analysis will beappropriate.

10Report writing
Candidates should be prepared to produce a wellordered, well-reasoned argument in a stylesuitable for a designated readership. (Thisreadership could be, for example, non-statisticalcolleagues, managers, or users of official reports.)
Candidates will be expected to make use ofgraphical methods to summarise data and identify
unusual features.

The use of technical words alone will not beenough; explanations must be helpful to the nonspecialist.

Interpretation of published data

Candidates should be able to provide criticalcomment on tables and graphs.

Graphs and diagrams must be clearly labelled sothat the reader understands them properly.

11

MODULE 2: Probability models
Probability
Definitions of probability: equally likely outcomes;
relative frequency; degrees of belief. Addition
and multiplication of probabilities, conditional
probability, statistical independence.
Bayes'
theorem.
Distributions
Random variables. Discrete and continuous
probability distributions.
Probability massfunction, probability density function, cumulative
distribution function.
Simple theory ofelementary probability distributions, including
discrete uniform, Bernoulli, binomial, Poisson,
geometric, negative binomial, hypergeometric,
Normal, exponential, gamma and continuous
uniform.

Derivation of cumulative distribution function fromprobability density function and vice versa.
Candidates should be able to show that theprobability mass function (probability density
function) of the named distributions sums
(integrates) to unity. Some knowledge of thederivation and application of these distributions is
expected. The Poisson approximation to thebinomial is included.
Candidates should know how these distributions
arise in practice and be able to recognise themfrom a brief description of a situation.

Properties of distributions
Expectation and variance; their general properties Derivation of the expected value and variance ofrandom variables with the distributions listed
and values for standard distributions.

above. Questions may be set for other simple
distributions.

Distributions, means and variances of sums of
independent and identically distributed randomvariables and simple functions, such as aX + b.
Linear combinations of independent Normally
distributed variables.

Results for distributions of sums of Poisson,
Normal and exponential random variables should
be known.
Distribution of ∑ ai X i when X i has N ( µi , σ i 2 )distribution.

Statement and use of central limit theorem for Proof of CLT not required.
independent, identically distributed randomvariables with finite variance.
Use of Normal approximations, including those Use of appropriate continuity corrections will be
for binomial and Poisson distributions.
expected.

12

MODULE 3: Basic statistical methods
Inference
Sample and population. Concept of a sampling Sampling distribution of the mean.
distribution. Standard error.
Point and interval estimates. Construction andinterpretation of confidence limits.
Hypothesis tests, test statistic, one- and two-sided Knowledge of p-values and their interpretation.
tests.
Significance level. Type I and II errors. Power as
1–P (type II error).
Use of Normal, t, χ2 and F distributions in testing Tests and confidence intervals involving means,variances and proportions. Use of tables toand interval estimation.
obtain percentage points.
Paired and unpaired two-sample tests.
Power curves.

Restricted to cases of testing for the mean of aNormal distribution with known variance.

The χ2 goodness-of-fit test of standard
distributions to observed data.

Including pooling of classes. Uniform (discreteand continuous), binomial, Poisson and Normal
distributions; distributions in specified
proportions.

Analysis of two-way contingency tables; χ2 test Use of Yates' correction in 2 × 2 tables isfor association.
expected.
McNemar’s test.

Use of a continuity correction is not expected.
Consideration of tied ranks will not be expected.

Non-parametric methods
Use of non-parametric and distribution-freesignificance tests for paired and unpaired data:
sign test, Wilcoxon rank sum test (Mann-Whitney
U test), Wilcoxon signed-rank test.

Candidates will be expected to be able to use
tables of percentage points but do not need toknow how the tables are obtained.
Consideration of tied ranks will not be expected.

13

MODULE 4: Linear models
Correlation
Product-moment correlation (Pearson). Rank Association versus causality. Simple tests using
correlation – Spearman’s coefficient. Calculation tables, with informal understanding of when eachmeasure of correlation is appropriate.
and interpretation.

Consideration of tied ranks will not be expected.

Design of experiments
Reasons for experimentation, causality.
Principles of replication and randomisation,completely randomised design.
Regression
Simple linear
estimation.

regression.

Leastsquares Models involving qualitative regressor variablesMultiple linear regression – concepts,
interpretation of computer output, inference forregression coefficients using estimates andestimated standard errors from computer output.

are not included in this module.

Knowledge of F test for regression and partial t
test for regression coefficients. Methods forselecting variables are not included in thismodule.

Analysis of variance for regression models.
Calculation and interpretation of the multiple R2 as a measure of the proportion of variation
correlation
coefficient(coefficientof explained.
determination).

Calculation of prediction intervals will not be
required.

Simple cases of transforming to linearity.

For example, problems (such as growth curves) inwhich taking logarithms or reciprocals leads to astraight-line relationship.

Analysis of variance
One-way analysis of variance.

Relationship to completely randomised design.

Inference for means and for differences in means.

Multiple comparison procedures will not be
required.

14

MODULE 5: Further probability and inference
Bivariate distributions
Simple bivariate discrete distributions. Joint, Distributions with probabilities presented either byconditionalandmarginal
distributions: formula or in a two-way table.
probability mass function, expectation andvariance. Covariance and correlation.
Simple bivariate continuous distributions. Joint, Simple cases only – rectangular and triangularconditionalandmarginal
distributions: spaces, with one or more sides parallel to aprobability density function, expectation and coordinate axis.
variance. Covariance and correlation.
The bivariate Normal distribution.

Familiarity with the bivariate Normal distribution
as a model (knowledge of joint pdf is not
required). Proofs of results are not required.

Generating functions

No limiting results.
Probability and moment generating functions.
Use to find expectations and variances. Use toestablish the distribution of sums of randomvariables.
Inference
The likelihood function.

Plots of likelihood against the parameter.

Estimation of a single parameter of a distribution Maximisation of likelihood using graphicalusing the method of moments and the method of methods and calculus. Questions may be set
maximum likelihood.
involving standard distributions listed in Module 2

and other simple cases.

Properties of point estimators.
Range, Introductory treatment of these concepts in simple
unbiasedness, consistency. Efficiency and relative cases.
efficiency.
Calculation of approximate variance of a Use in constructing a confidence interval.
maximum likelihood estimator using second Regularity conditions may be assumed.
derivative of log likelihood.

15

MODULE 6: Further applications of statistics
Design and analysis of experiments
Principles of design including randomisation, Reasons for using these.
blinding, pairing and blocking.
Randomised block designs. Latin squares.

Analysis of variance, inference for means and for
differences in means.

Factorial treatment structure with two factors.
Advantages of factorial experimentation.
Diagrammatic explanation of interaction.
Two-way analysis of variance.

Diagrams of means of treatment combinations
and their use for explaining interactions of two
factors.
Analysis of variance, inference for means and for
differences in means.

Residuals and their use in checking assumptions.
Multiple regression
Least squares estimation for multiple regression.

Extension of material in Module 4. Derivation of
normal equations in simple cases. Matrix
notation will not be required. Solution ofsimultaneous equations in simple cases only.

Regression through the origin.

Including multiple regression with zero intercept.

Use of backwards elimination in multiple Use of F tests.
regression.
Polynomial regression.

Simple cases only.

Use of indicator variables to model factors or Simple cases only.
qualitative variables.
Residuals and their use in checking assumptions.
Quality control and acceptance sampling
Charts for mean and range for Normal data. Construction and use of Shewhart charts,
including use of warning and action lines.
Charts for proportions.
Cusum charts.

Construction and use.

Attribute sampling. Single and double sampling
schemes.

16

MODULE 7: Time Series and Index Numbers
Time series
Decomposition of time series into trend, cycles, Additive, multiplicative
seasonal variation and residual (irregular) decompositions.
variation.

and pseudo-additive

Estimation of trend using regression or moving Equally and unequally weighted moving averages.
averages.
Examination of seasonal terms in time series Prior adjustment. Detection and control of special
decompositions. Seasonal adjustment using events in seasonal adjustment, for example
regression or moving averages.
religious holidays, changes in taxation levels, … .
Elementary forecasting methods: exponential
smoothing and Holt-Winters.
Introduction to ARIMA models.

Definition, interpretation, forecasting. Details oftechniques for fitting ARIMA models are notrequired.

Examination and interpretation of residuals from Including the correlogram and the portmanteau
fitted models.
lack-of-fit test.
Interpretation of computer output.

Analysis and commentary on tables and graphs
produced by seasonal adjustment programs.
Confirmation of some results through handcalculation.

Index numbers
Introduction to index numbers
Index numbers and their uses. Simple pricerelatives, Laspeyres and Paasche. Relationshipbetween Laspeyres and Paasche; and the relative
merits of each. Further index numbers –Törnqvist, Walsh, Fisher and Geometric Laspeyres.
Index aggregation.

Consideration of price and volume indices, with
examples from candidate’s knowledge andexperience.
Calculation of index numbers by aggregatingothers.

Deflation
Why deflation is used and how it works; what With reference to specific examples frommakes a good deflator; how deflation is carried candidate’s knowledge and experience.
out.
Re-basing
Why, when and how re-basing is done.

With reference to specific examples fromcandidate’s knowledge and experience. Benefits
and pitfalls.

Chain linking
Chain linking of simple price relatives, and chain With reference to specific examples fromcandidate’s knowledge and experience. Benefits
linking using Laspeyres.

and pitfalls.

Use of index numbers

With reference to specific examples fromcandidate’s knowledge and experience, forexample within National Accounts.
17

MODULE 8: Survey Sampling and Estimation
Populations and frames
Target and study populations. Types of framesavailable, uses and sources of error.
Sampling methods
Non-probability methods, haphazard sampling, Advantages and drawbacks of these methods,
sources of bias, unknown precision.
quota sampling.
Simple random sampling, stratified random Uses, benefits and limitations of each method.
sampling (with equal, proportional and optimal Discussion of practical and theoretical utility ofallocation), cluster and multi-stage sampling, methods in the context of specific examples.
systematic sampling.
Simple and stratified random sampling
Uses, limitations, applications to different datatypes, practical examples.
Estimates of totals, means and proportions,construction of confidence intervals.
Neyman and optimal allocation, use to reduce Discussion and comparison of these methods.
variance.
Calibration techniques for estimation
Ratio and regression methods.
supplementary information.
Practical problems
conducting surveysin

Use of Discussion of how and why other survey

planning

information may be useful in estimation.
Formulae will not be required.

andChoice of sampling method and estimators to be Cross-sectional, longitudinal and panel surveys,
used in a survey, trade-off between bias and drug trials, etc.
variance. Discussion of sampling problems in an
actual survey, recommendations for improvement
using practical examples.

18

Advised academic progression
through the modules
The Society does not operate any system of formal prerequisites for the modules. Any candidate
may enter for any module at any time. However, we strongly encourage candidates to ensurethat they are properly prepared for the modules they intend to take. As advised earlier, entry to
any of the modules assumes knowledge of the material in the Ordinary Certificate in Statistics, or
equivalent. Candidates should therefore ensure that they are familiar with this material before
sitting for any modules. This is particularly important for candidates who wish to attempt toachieve the Higher Certificate in Statistics by taking six modules (including modules 1 to 4) in thesame session. Such familiarity with the material need not have been gained by formally passing
the Ordinary Certificate. It could have been achieved in other ways, including by experiential
learning through a candidate's work.
Candidates are strongly advised to study the published specimen papers, past papers andsolutions. These give a good indication of the sorts of questions that will be asked, their generallevel, and the breadth and depth of answers that will be expected.
Although there are no formal prerequisites, it is an inherent feature of the subject that somemodules require knowledge of some topics in other modules. Candidates may of course havealready acquired such knowledge through other routes, or may be willing to acquire it as part oftheir studies for the module(s) at which they are aiming. As guidance, the Society offers thefollowing advice, but this should not be seen as exhaustive:
Module
1
2
3
4
5
6
7
8

Some dependence on topics in Module or Modules
–
–
2
2, 3
2, 3, 4
1, 2, 3, 4
1, 2, 4
1, 2, 3, 4

The form and extent of any dependence varies considerably from case to case. For somemodules, detailed knowledge of some of the topics contained in other module(s) might beneeded. In other cases, candidates will just need to be familiar with or have some knowledge ofthe concepts and techniques contained in the other module(s).

19Mathematics Background
The mathematics required to support the modules in the Higher Certificate in Statistics isnaturally at a higher level than that required for the Society's Ordinary Certificate. A summary of
the required mathematics is set out here, but prospective candidates should study the publishedspecimen and past papers and solutions to get a detailed understanding of what is required ineach module.
The examination papers will not concentrate on mathematics for its own sake. Rather, it is theability to apply mathematics within the statistical contexts defined by the module syllabuses thatwill be examined.
Candidates should be aware that the general level of the Higher Certificate is, broadly, that of thefirst year of a university undergraduate degree in Statistics, also including some topics that mighttypically be found in the second year. The level of mathematics required is commensurate with
this.
The detailed mathematical requirements naturally vary from module to module. As a general
guide, a candidate who has knowledge of the topics contained in A-level Mathematics in theschools examination system in England, or the equivalent in other systems, will certainly be able
to start work on any of the modules. This does not necessarily mean that A-level Mathematics(or equivalent) has been formally taken and passed (or been studied concurrently). It is intended
as a guide to the breadth and depth of mathematical topics with which candidates should befamiliar. Some of the modules do not in fact require as much mathematics as this. On the other
hand, completion of the work in other modules will require knowledge of additional
mathematical topics beyond A-level Mathematics (or equivalent).
The table on the next page sets out the various mathematical topics in broad headings andindicates which are required for which modules.

Interpretation of Computer Output
In any of the modules, candidates will be expected to be able to interpret computer output fromstatistical packages. Detailed knowledge of specific packages is not required.

20

Table showing mathematics requirements for each module(see notes on previous page)

Topic
AlgebraUse of Σ notation
Permutations and combinations
Solutions of linear and quadratic equations
Manipulation and solution of simple inequalities
Arithmetic and geometric series
Summation of series
Limits of sequences and functions
Positive numbers raised to any real power. Exponential
and logarithmic functions, including their expansions inseriesUse of the following results:lim(1+ n – 1 ) n = e and lim(1+( x / n )) n = e x
Use of binomial theorem with any integer index
Solution of simple sets of linear equations having uniquesolutionsDifferential calculus
Derivatives of polynomial, logarithmic and exponentialfunctions, and of sums, products, quotients or functions ofthese functions
Maxima and minima. Simple examples of asymptotes
Graphical representation of functions and simple examples
of curve sketching
Simple examples of partial differentiation
Integral calculus
Indefinite and definite integrals, including those with
infinite limits
Integrals of algebraic, exponential and logarithmic
functions
Simple examples of integration by substitution and byparts, including reduction formulae
Double integration – elementary aspects only

1

2

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Graduate Diploma in Statistics
The Graduate Diploma of the Royal Statistical Society is offered in a modular form. This gives theopportunity for candidates to proceed at their own pace. The Graduate Diploma as a whole is aqualification in applied statistics at a level equivalent to that of a good UK Honours Degree in
Statistics. In addition, the individual modules are intended as valuable free-standing elements intheir own right; they may for example be useful components of a formal or informal continuing
professional development programme.
Candidates may enter for any number of modules, either singly or in combination, in any
examination session. Candidates may resit any module any number of times; the highest markachieved will stand. The pass mark for each individual module is 50%. Module marks are"banked" without time limitation.
Each module is examined by a 3-hour written paper containing eight questions, of which
candidates are asked to answer five. Candidates will be advised of their marks for each moduletaken. Candidates will receive a separate certificate for each module in which they aresuccessful. Candidates who are successful in all five modules (not necessarily all in the samesession) will be awarded the Graduate Diploma in Statistics. The Graduate Diploma will only beawarded on the first occasion when a candidate becomes eligible for it. If a candidate, having
been awarded the Graduate Diploma, takes any module(s) again, advice of the marks earned will
be issued and, if appropriate, a certificate to indicate success in the individual module(s); but theGraduate Diploma itself will not be re-awarded.
Candidates are advised that entry to any of the modules assumes knowledge of the material inmodules 1 to 6 (inclusive) of the Higher Certificate in Statistics, or equivalent (note: the former
non-modular form of the Higher Certificate in Statistics, offered for the last time in 2008, isequivalent). Questions may be set using this material, whether or not a topic is specificallymentioned in the individual Graduate Diploma module syllabuses below.
Candidates must not regard the Graduate Diploma modules as being fully academically separate
from each other, because there are many interactions between branches of the subject at thislevel. In all cases, study of one module may require some study of work in other modules. This isespecially true of modules 4 and 5, but it is reiterated that it may apply to all modules. For thisreason, and because candidates are likely to have particular interests and particular strengths inparticular areas, the Society does not suggest an advised academic progression through themodules in the manner that it does for the Higher Certificate. To some extent modules 1 and 2
can be considered as a pair and modules 4 and 5 as another pair, with these pairs being
independent of each other and with module 3 being more closely related to modules 1 and 2
than it is to modules 4 and 5. However, this should not be viewed as a strict separation; there
are potential connections between all of the modules.
A summary of the mathematics background required for the Graduate Diploma is set out below,
but prospective candidates should study the published specimen papers, past papers andsolutions to get a detailed understanding of what is required in each module. The mathematics
background is naturally at a higher level than that required for the Higher Certificate. It iscommensurate with what might typically be required for the final year of an undergraduate
degree in Statistics in a UK university. The Society does not separate the required mathematics
background into distinct sections for each module. Candidates should assume that they mightneed any of the required mathematics in any of the modules.
22

The examination papers will not concentrate on mathematics for its own sake. Rather, it is theability to apply mathematics within the statistical contexts defined by the module syllabuses thatwill be examined.
In any of the modules, candidates will be expected to be able to interpret computer output fromstatistical packages. Detailed knowledge of specific packages is not required.
Candidates are strongly advised to study the published specimen papers, past papers andsolutions. These give a good indication of the sorts of questions that will be asked, their generallevel, and the breadth and depth of answers that will be expected.

Module Syllabuses
Syllabuses for the five modules are set out on pages 25 to 30.
The modules are
Module 1

Probability distributions

Module 2

Statistical inference

Module 3

Stochastic processes and time series

Module 4

Modelling experimental data

Module 5

Topics in applied statistics

23

MODULE 1: Probability distributions
This module concentrates on probability and distribution theory, extending the coverage of these topics in the Higher
Certificate. Simulation is also included.

Probability
Sampling with and without replacement.
Elementary problems involving urn models.
Joint probability, marginal and conditional
probability, independence.
Law of total probability. Bayes' Theorem.
Distribution theory
Random variables. Discrete and continuous
random variables. The probability mass function
and probability density function. Cumulative
distribution function.
Expectation as a linear operator. Expectation offunctions of a random variable. Mean andvariance.
Approximate mean and variance of a function of Derivation ofa random variable.
Variance-stabilising approximation.
transform-ations.
transformations.
Standard distributions and their use in modelling,including Bernoulli, binomial, Poisson, geometric,
negative binomial, hypergeometric, discrete
uniform, Normal, exponential, gamma, continuous uniform, beta, Weibull, Cauchy, lognormal.

E[f(X)] by Taylor series
Use in finding appropriate

Ability to recognise the appropriate distribution
from a model description.
Relationships between distributions (e.g.
exponential and Weibull).

Joint, marginal and conditional distributions.
Independence. Covariance and correlation.
The multivariate Normal distribution. Regression; E(Y | X=x) and E(X | Y=y).
multiple correlation, partial correlation.
Knowledge and use of E ( Y ) = E ( E ( Y | X )). Proofs not required. Recall of the result for Var(Y)Use of Var( Y ) = E (Var( Y | X )) + will not be expected.
Var( E ( Y | X )).
Probability generating function.
Momentgenerating function. Applications of generating
functions. Distribution of sums of random Sums of fixed and random numbers of randomvariables, and of sample mean.
variables.
Central limit theorem.

Proof for independent and identically distributed
random variables only.

Distributions of functions of several random Univariate and bivariate transformations.
variables.
Transformations, including theprobability integral transform. Joint distributionof mean and variance from a Normal random
sample.
The t, χ2 and F distributions, and their use as Including definitions in terms of N(0,1) randomvariables.
sampling distributions.

24

Joint distribution of order statistics. Distributionof sample range.
Simulation
Generation of uniform pseudo-random numbers;
testing for uniformity.
Methods of generating random numbers from Applications of methods to standard distributions
common distributions, including inversion, including binomial, Poisson, exponential, Normal,
rejection and table look-up techniques. Monte logistic.
Carlo methods. Use of variance reduction
techniques. Applications of simulation.

25

MODULE 2: Statistical inferenceThis module extends the coverage of the material on statistical inference in the Higher Certificate. It also introduces
Bayesian inference, decision theory and non-parametric inference.

Estimation
Unbiasedness, mean square error, consistency,
relative efficiency, sufficiency, minimum variance.
Fisher information for a function of a parameter,
Cramér-Rao lower bound, efficiency. Fitting
standard distributions to discrete and continuous
data. Method of moments. Maximum likelihood
estimation: finding estimators analyt-ically andnumerically, invariance.
Hypothesis testing
Simple and composite hypotheses, types of error,
power, operating characteristic curves, p-value.
Neyman-Pearson method. Generalised likelihood
ratio test.
Use of asymptotic results to construct tests.
Central limit theorem, asymptotic distributions ofmaximum likelihood estimator and generalised
likelihood ratio test statistic.

Proof of the asymptotic distributions of the
maximum likelihood estimator and thegeneralised likelihood ratio test statistic are notrequired.

Confidence intervals and sets
Random intervals and sets. Use of pivotal
quantities.
Relationship between tests andconfidence intervals. Use of asymptotic results.
Data resampling
Reduction of bias. Estimation of precision. Jack-knifing and bootstrapping.
Confidence interval estimation.
Non-parametric inference
Permutation and randomisation tests.
ranks and randomisation; robustness.

Use of Sign, Wilcoxon rank sum (Mann-Whitney U),

Wilcoxon signed-rank, Kolmogorov-Smirnov (oneand two samples), goodness-of-fit and rankcorrelation tests.

Bayesian inference
Prior and posterior distributions. Choice of prior:bets, conjugate families of distributions, vague
and improper priors. Predictive distributions.
Bayesian estimates and intervals for parametersand predictions. Bayes factors and implications
for hypothesis tests. Use of Monte Carlo Knowledge of computationally intensive methods
simulation of the posterior distribution to draw for simulating posterior distributions (e.g. Markov
inferences.
Chain Monte Carlo) is not required.
Decision theory
Loss, risk, admissible and inadmissible decisions,randomised decisions. Minimax decisions andBayes' solutions, including simple results.

26

Comparative inference
Different criteria for choosing good estimators,
tests and confidence intervals.
Different
approaches to inference, including classical,
Bayesian and non-parametric.

27

MODULE 3: Stochastic processes and time seriesThis module provides an extended coverage of stochastic processes, including Markov chains and various forms ofPoisson processes, and of time series, including ARIMA modelling.
The syllabus concentrates on underlying theory, but applications in various substantive areas are also important andwill be represented in examination questions.

Stochastic processes
General stochastic process models.
Random walks.
Reflecting and absorbing barriers.
Mean recurrence time, mean time to absorption.
Difference equations.
Branching processes.

Use of generating functions.
Recurrence relations for size of n th generation;
probability of extinction.

Markov chain models for discrete-state processes. Not restricted to finitely many states.
Transition matrices: 1-step and n-step.
Classification of states.
Equilibrium distributions for time-homogeneous
chains.
Poisson processes.
Differential-difference equations.
Birth and death processes.
Queues.
The M/M/1 queue. Differential-difference equations. Conditions for equilibrium. Equilibrium Equilibrium behaviour for queues with transition
distributions of queue size and waiting time for rates dependent on queue size.
first-come-first-served queues. Extensions to
M/M/k and M/M/ ∞ queues.
The M/G/1 queue, imbedded Markov chain
analysis.
The Pollaczek-Khintchine formula. Equilibrium treatment only.
Mean queue length and waiting time.

Time series
Time series models; trend and seasonality.

In examination questions, the word "queue" willrefer to all units in a system, i.e. those being
served as well as those still waiting to be served.
Additive and multiplicative models.

Stationarity.
Autocovariance, autocorrelation and partial
autocorrelation functions. Correlograms.
Autoregressive (AR) processes.
Yule-Walker equations.
Moving average (MA) processes.
Invertibility conditions.
ARMA processes.
ARIMA processes and Box-Jenkins methods.
Identification, estimation, checking, forecasting.

Box-Pierce and Ljung-Box statistics.

Forecasting and minimising expected prediction Exponential smoothing, Holt-Winters.
variance.
Introduction to frequency domain analysis. Candidates will be expected to have someSpectral density function. Periodograms.
familiarity with the fast Fourier transform.
28

MODULE 4: Modelling experimental data
This module covers the general and generalised linear model. It also covers the design of experiments and analysis ofexperimental results using analysis of variance and multiple regression, extending the coverage of these topics in theHigher Certificate. Candidates are expected to have knowledge of the use of these methods and the interpretation of
results from them, including computer output: critical evaluation of computer output is required throughout thismodule.

General linear model
Least squares. Properties of LS estimators in thelinear model, Gauss-Markov theorem. Models for
simple and multiple regression and for analysis ofvariance and covariance. Hat matrix. Estimationof variance. Interval estimates of parameters.
Weighted least squares.
Importance ofassumptions. Analysis of residuals. Transformation of variables. Linearising other models, e.g.
multiplicative models and growth curves.
Design of experiments
Randomisation, replication, blocking.
Completely randomised designs, randomised
blocks, Latin squares, balanced incomplete
blocks.
Factorial treatment structure.
2n designs, including confounding and fractional
replication.
Analysis of variance
Analysis of variance for the designs listed above,
including for cross-classifications with replication, and for nested or hierarchical designs.
Fixed and random effects; variance components.
Application to data collected in experiments or bysampling.
General linear contrasts among treatments.
Inference for individual treatment means and forcontrasts.
Analysis of residuals. Use of plotting techniques
to detect non-Normality of errors.
Multiple regression
Regression with more than one explanatory
variable. Use of indicator variables to represent
factors. Analysis of variance of regression,
including test for lack of fit. Analysis of residuals,
regression diagnostics, detection of influential
observations, multicollinearity, serial correlation.
Model selection, using all-subset and stepwise
methods.
Extension to non-linear modelling.
Fitting
standard growth curves (including logistic andGompertz). Estimation of parameters, approximate confidence intervals and tests.

Proof of standard results for multiple regression(matrix notation; matrix differentiation will not
be required). Application of these to othersituations as listed.

Reasons for using these designs; bias andprecision. How to construct a valid randomisedlayout for each design. Knowledge of whatbalanced incomplete block designs exist forreasonably small block sizes and numbers oftreatments.

Estimation of variance components, and use inplanning sampling schemes.
Comparisons between and within groups oftreatments. Linear and quadratic components forfactors at equally spaced levels.

The Extra Sum of Squares principle.
Pure error.
Concept of leverage. R2 and adjusted R2. Use of
the Durbin-Watson statistic.
Including use of Cp plots.
Candidates will be expected to have somefamiliarity with the Newton-Raphson procedurefor fitting non-linear models.

29

Generalised Linear Model
Theory and examples of use. Link functions.
Logistic regression for analysis of binary data.
The multinomial distribution. Log-linear models
for analysis of cross-tabulated categorical data.
Deviance; analysis of deviance. Use of F test forcomparing models. Diagnostics.

Poisson regression. Components of a generalised
linear model including the exponential family.
Adapting to various problems by choosing
suitable link functions etc. Candidates will be
expected to have some familiarity with themethod of iterative reweighted least squares usedin fitting models.

30

MODULE 5: Topics in applied statistics
This module covers four application areas, denoted by bold sub-headings below. It introduces some generaltechniques for data analysis and some that are more specialised to particular areas. The paper for this module will
not be formally divided into sections but it will always consist of two questions on each of the four application areas
covered by the syllabus. Candidates are particularly advised that some questions on the paper are likely to require
knowledge of material covered in the module on Modelling Experimental Data (module 4).
Critical evaluation of computer output is required throughout this module. An appreciation of the problems raised by
missing values in data is expected throughout this module.
Note that this syllabus is continued on the next page.

Multivariate methods
Vectors of expected values. Covariance andcorrelation matrices.
Discriminant analysis, choice between two Linear and quadratic discrimination.
populations, calculation of discriminant function,and probability of misclassification, test andtraining samples, leave-one-out and k-fold crossvalidation, idea of extension to several
populations.
Principal components; definition, interpretation of Based on covariance and correlation matrices.
calculated components, use in regression.
Cluster analysis, similarity measures, single-link
and other hierarchical methods, k-means.
Informal approaches to checking for multivariate
Normality. Tests and confidence regions for Hotelling's T 2.
multivariate means.
Censored data: survival and reliability
Problems involving censored data, for example inclinical and engineering contexts.
Reliability and life testing.
Hazard and survivor functions.
Confidence intervals for survivor function using
Kaplan-Meier estimate of survivor function.
Greenwood's formula.
Weibull and hazard plots.
Logrank test.
Parametric survival distributions - exponential, Use of log cumulative hazard plot to check
Weibull.
Weibull and proportional hazards assumptions.

Use of these distributions in modelling survival
data. Fitting methods not required.

Proportional hazards and Cox regression.

An understanding of the assumptions andinterpretation of the fitted model. Details ofpartial likelihood and numerical methods forfitting the model not required. Calculation andinterpretation of hazard ratios and confidence
intervals.

Checking for non-proportionality of hazards.

Checking for non-proportional hazards using alog cumulative hazard plot and plots of hazard
functions.

31Demography and epidemiology
Population pyramids.
Life tables.
Standardised rates (e.g. mortality).
Incidence and prevalence.
Design and analysis of cohort (prospective)
studies.
Design and analysis of case-control (retrospective) studies.
Confounding and interaction.
Matched case control design and analyses, using
McNemar's test.
Causation.
Relative risk. Odds ratio. Estimation andconfidence intervals for 2×2 tables.
Mantel-Haenszel procedure.
Sensitivity, specificity, ROC curves, positive
predictive value, negative predictive value.

Construction and use of life tables; derived
quantities, including calculation of life
expectancy. Direct and indirect standardisation.
Distinction between these concepts.

Reasons for matching; advantages and disadvantages relative to unmatched studies.
Inferring causality from observational studies.
Use in adjusting for confounding variables.
Uses in screening and diagnosis.

Sampling
Census and sample survey design. Target and Revision and extension of basic concepts fromstudy populations, uses and limitations of non- Higher Certificate.
probability sampling methods, sampling frames,
sampling fraction.
Simple random sampling. Estimators of totals, Examples of practical use in various contexts.
means and proportions;bias.
Estimated
standard errors, confidence intervals andprecision. Sampling fraction and finite population
correction. Ratio and regression estimators.
Use of supplementary information.
Stratified random sampling. Estimators of totals, Examples of practical use in various contexts.
means and proportions;bias.
Estimated
standard errors, confidence intervals andprecision. Cost functions. Proportional and Minimisation of cost × variance.
optimal allocations. Limitations of stratified
sampling.
One-stage cluster sampling. Estimators for totals, Examples of practical use in various contexts.
means and proportions with equal cluster sizes
and with different cluster sizes. Estimatedstandard errors, confidence intervals andprecision.
Link with systematic sampling.
Description of two-stage sampling and of multistage sampling. Limitations.

32Mathematics Background
Knowledge of the mathematical topics listed below is required. These include those topics previously
specified for the Higher Certificate (see page 20). The ability to apply this mathematics will be examinedwithin the statistical contexts defined by the module syllabuses.
AlgebraPermutations and combinations. Partial fractions, including quadratic factors. Solution of linear andquadratic equations. Manipulations and solution of simple inequalities. Trigonometric functions andtheir inverses. Summation of series with Σ notation. Limits of sequences and functions. Geometricseries. Exponential and logarithmic functions, including their expansions in series and the results
n

n
 x
e x . Taylor series expansion. Double summation (ΣiΣj).
e and lim 1 +  =lim (1 + n −1 ) =
n →∞
n →∞
n



Positive numbers raised to any real power. Use of the binomial theorem with any real index.
Differential calculus
Differentiation. Derivatives of polynomial, trigonometric, inverse trigonometric, logarithmic andexponential functions, and of sums, products, quotients or functions of these functions. Maxima andminima; asymptotes; points of inflexion (defined as those points at which a curve crosses its tangent).
Graphical representation of functions and simple examples of curve sketching. Partial differentiation.
Integral calculus
Integration. Indefinite and definite integrals, including those with infinite limits. Integrals of algebraic,
exponential, logarithmic and trigonometric functions. Simple examples of integration by substitution andby parts, including reduction formulae.
Double integration over rectangular and triangular regions having one or more sides parallel to a coordinate axis. Functions of a single variable and of two variables. Interchange of order of integration.
Jacobians of transformations.
Matrix algebra
Vectors. Square matrices: symmetry, singularity and non-singularity, determinants, inverses, relation
with sets of linear equations. Solution of simple sets of linear equations.
Rectangular matrices: addition, subtraction and multiplication. Rank of matrices via linear dependence
and independence of vectors. Quadratic forms, including their expression in matrix notation.
Numerical methods
Iterative solution of equations, including Newton-Raphson method.

33