7 Post-16 provision
7.1 A level Mathematics
All the syllabuses for AS and A level Mathematics, and for Further Mathematics, are modular; three modules are needed for AS level and six for A level. Most of the syllabuses have four statistics modules, which are cumulative in content; thus Statistics 4 assumes knowledge of Statistics 3 which assumes Statistics 2 which in turn assumes Statistics 1.
At AS level, two of the three modules must be pure mathematics leaving room for only one applied module, which may be (and often is) Statistics 1. (The other options for the applied module are Mechanics 1 and Decision Mathematics 1.)
Similarly, for A level four of the six modules must be pure mathematics and only two are applied; these may be Statistics 1 and 2, or Statistics 1 and a module from another strand, for example Mechanics 1, or they may both be from other strands.
Statistics 3 and 4 inevitably belong to Further Mathematics.
For a number of reasons, available data do not allow an exact determination of the number of people in any cohort who take the various statistics modules; however, the number of students taking AS and A levels in Mathematics is subject to quite large changes (currently increases) from year to year so there would be limited value in such a figure anyway. The best that can reasonably be done is quite a rough estimate.
This is given in Table 9 for the cohort who took their A levels in 2011.
Qualification Estimated numbers
AS level Mathematics 110,000
A level Mathematics 83,000
Statistics 1 100,000
Statistics 1 & 2 30,000
Statistics 1, 2 & 3 3000
Statistics 1, 2, 3 & 4 250
Table 9 Estimated numbers taking Statistics modules within AS & A level Mathematics and Further Mathematics in 2011
The content of the statistics modules varies between syllabuses. Table 10 lists typical content for Statistics 1 and 2. Items in brackets are in some but not all of the syllabuses.
Topic Notes
Data display Revision of GCSE Higher tier
Data measures Revision of GCSE, Standard deviation
Random variables Discrete, (Continuous)
Distributions Uniform, Binomial, Poisson, Normal
(Approximations)
Bivariate data Correlation, Regression
Central Limit Theorem (Confidence intervals)
Hypothesis testing Binomial, Normal, (Poisson), (t-test), (Correlation), (Ȥ2)
Probability Revision of GCSE Higher tier and extension
to conditional probability Table 10 Typical content of Statistics 1 and 2 modules
During the 1990s some A level Mathematics syllabuses included statistics
coursework, either as an option or as a mandatory part of the assessment. So at that time, there were some mathematics students who encountered this aspect of
statistics; for many of them, this was a rich and valuable experience. In 2000 revised
Paradoxically, this was undermined by the introduction of statistics coursework into GCSE Mathematics in 2003. For reasons that have already been described, it became common for GCSE students to have a poor experience with their statistics coursework and so to be reluctant to take on similar work the following year as part of AS level Mathematics. Statistics coursework acquired such a bad name that it was withdrawn from AS level or became optional.
Virtually all of the statistics in AS and A level Mathematics syllabuses is now assessed entirely by timed written examinations, and so there is no requirement to design or carry out an experiment. (A partial exception occurs with Statistics 1 in the AQA syllabus which allows candidates to replace an examination question by
coursework. Few candidates, less than 2%, take this option.)
A few teachers may, as a matter of good practice, require their students to do such work but they are exceptional. By far the majority of students taking AS or A level Mathematics derive no practical experience of designing or carrying out an
experiment from it, or of deciding how to process their own data. They learn to answer examination questions which require them to use particular techniques to analyse small data sets. A typical question is shown in Example 7.
The random variable X represents the reaction times, in milliseconds, of men in a driving simulator.
X is Normally distributed with mean 355 and standard deviation 52.
(i) Find
(A) P(X < 325),
(B) P(300 < X < 400). [6]
(ii) Find the value of k for which P(X < k) = 0.2. [3]
It is thought that women may have a different mean reaction time from men. In order to test this, a random sample of 25 women is selected. The mean reaction time of these women in the driving simulator iVPLOOLVHFRQGV<RXPD\DVVXPHWKDWZRPHQ¶VUHDFWLRQWLPHVDUHDOVR1RUPDOO\
distributed with standard deviation 52 milliseconds. A hypothesis test is carried out to investigate whether women have a different mean reaction time from men.
(iii) Carry out the test at the 5% significance level. [8]
Example 7 A typical statistics question in A level Mathematics24
6JGTGKUPQVJKPIKPVTKPUKECNN[YTQPIYKVJSWGUVKQPUNKMGVJKU6JG[CUUGUUUVWFGPVUŏ knowledge of the techniques in the syllabus efficiently, and suggest contexts in which they could be used. Such questions will always be needed but they do not ensure that students meet the full statistics cycle. This does not happen at present, and so the experience of students taking AS and A level Mathematics is summarised by Figure 7.
Figure 7 Statistics in AS and A level Mathematics
In one mathematics syllabus25, all A level candidates are required to take a comprehension paper. This often requires candidates to read and understand an example of quite detailed problem analysis. Many of those set have been statistical in nature; titles of some of them are given in Example 8.
Comprehension topics Card shuffling
Screening for rare diseases 7UDLQV¶SXQFWXDOLW\
Directional data
Herd immunity
Modelling athletics records Credit card fraud
Estimating animal populations
Example 8 A sample of comprehension topics used in recent years26
While most students taking A level Mathematics do not progress beyond Statistics 1, there are those who take Statistics 2 as well and some Further Mathematics
students go on to Statistics 3 and 4. The emphases of the various examination boardsŏ syllabuses reflect somewhat different underlying philosophies. Nonetheless, many topics are to be found in all of them. The statistics available overall is
summarised in Table 11.
In 2004, the requirements of A level Mathematics were reduced from three applied modules to the present two. The effect has been to reduce the amount of statistics many students learn, and also what is available; until then, many syllabuses had six statistics modules but this has now been reduced to four. (This change was
designed to rectify the problems caused by Curriculum 2000 in which mathematics was seen to be too demanding, with a loss of nearly 20% of A level candidates.)
Problem analysis
Data presentation
Data collection Data analysis
Data Hypothesis tests Measures of location Language of hypothesis tests
Measures of spread Type 1 & 2 errors
Data presentation Binomial test for probability
Samples Normal test for mean
Sampling t-test for mean
Samples and populations Ȥ2 test (contingency table) Probability Ȥ2 test (goodness of fit) Laws of probability Test for Poisson parameter Conditional probability Test for correlation
Permutations and combinations Paired sample test(s) Random variables Two-sample test(s) Discrete random variables Wilcoxon test(s) Continuous random variables Tests for variance Linear combinations of random variables Analysis of variance
Bivariate data Power of a test
Correlation Estimation and confidence intervals
Linear regression Estimation
Rank correlation Confidence interval for mean
Distributions Confidence interval for variance Uniform (rectangular) distribution Other techniques
Binomial distribution Estimators
Poisson distribution Generating functions
Normal distribution Exponential distribution
Geometric distribution Key
Discrete bivariate distributions In five or six of the six syllabuses Central Limit Theorem In three or four of the six syllabuses Distributional approximations In one or two of the six syllabuses
Table 11 Statistics topics in the A level Mathematics and Further Mathematics syllabuses