RSSCSE

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Statistical Awareness
Citizen's knowledge
Levels of understanding
A Programme of Study
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Here you will find our proposals for the extent, level and form of awareness and knowledge of statistics that every citizen should have. Implicit in the proposals is that the material should be taught through a problem solving approach, similar to the paradigm followed by practising statisticians or people carrying out scientific inquiry.

We describe three levels of knowledge and awareness: foundation; intermediate; and advanced. The foundation level would be roughly equivalent to what is currently studied in statistics by UK learners by the age of 14. Intermediate level corresponds to what is studied by the age of 16 (UK GCSE Mathematics), while Advanced level corresponds to what learners study by the age of 19 (content of some UK GCE ‘A’ level Mathematics, Statistics or other equivalent subjects).

1 Components of Statistical Education

1. The diagram here emphasises our belief that all students should be made statistically aware through a core that teaches a strategy for learning statistics through an iterative cycle of activities. These are represented by the five (yellow) rectangular boxes in the centre of the diagram.
2. The next outer ring of boxes broadly divides topics into three areas of application, scientific, official data and surveys. A Foundation level qualification is represented by the (brief) outline of content in the first continuous rectangular band of statistical literacies.
3. The second rectangular band represents Intermediate and the outer band Advanced.

This diagram emphasises the nature of the statistics that students should be able to do at the various levels. At each level they should know about the techniques and sort of questions that might be asked at the next level and about the more complex contexts suitable for the next level.

As you move out from Foundation through Intermediate to Advanced:

1. the contexts will become more complicated and more specialised;
2. the techniques introduced at one level will become used in more complex contexts and tasks at the next level.

Every student that leaves school should have the knowledge and skills defined through using the core statistics problem solving paradigm together with the Intermediate Level material briefly described within the second outer rectangular band of statistical literacy. These should be part of their common knowledge skills and attributes for entering work. In section 2 we expand on the ideas behind the proposal. In section 3 we define the levels of understanding that should be acquired for intermediate statistical education.

2 What all citizens should know

The emphasis of nearly all school statistics courses has been on getting students to do statistics. A level 2 statistics course that all students who leave school ought to have studied should also include material on what statistics (or statisticians) can do. That is, there should be material on how statistics is used in society, the sort of questions that statisticians answer and how those answers can be interpreted. A well-educated school leaver should know about some of the success stories in statistics and the current areas in which statisticians are making a contribution. They should be aware of the many and various contexts in which statisticians have and do work – in many aspects of real life as well as in many different academic subjects. They should be helped to understand both the strengths and the limitations of the statistical way of approaching issues.

1. Statistics is now so wide-ranging a subject that it is difficult to define. You can try to infer what it is by looking at what statisticians do. On the Royal Statistical Society’s web site the following phrases can be found to describe different aspects of statistics.
2. Statistics changes numbers into information.
3. Statistics is the art & science of deciding what the appropriate data to collect are, deciding how to collect them efficiently and then using them to give information, answer questions, draw inferences and make decisions.
4. Statistics uses the language & ideas of probability to describe inferences and risk.
5. Statistics uses samples to get insight into different populations.
6. Statistics is making decisions when there is uncertainty.

In the UK there are currently two GCSE courses in Statistics available – from AQA and from Edexcel. Both of these specifications include an aim that students should acquire an understanding of the basic concepts of probability and statistics in such a way which encourages confidence satisfaction and enjoyment of the subject in everyday situations familiar to the student and in other disciplines. They also have an aim that students should develop an awareness of the importance and limitations of statistical information to society as a whole. These seem appropriate as major aims for a basic level 2 statistical education for all school leavers. It is unfortunate that these two aims are largely under-represented in the assessments set for qualifications at this level.

Current (UK) school courses in statistics concentrate on the process implied by point 4. above. This is a major part of the way statisticians think and should not be lost. However, the practical problems of getting good samples from populations should be part of any basic statistical education course. This also includes the fact that such data may be obtained by experiment or by survey and the need to obtain good and accurate information (with its implications for questionnaire design) is part of this. There is, however, a growing amount of statistical work that does not come easily under this population – sample structure. At one extreme there is the National Census of Population where the whole population (in theory) is measured. But there are also many areas where the data are there because they are collected routinely – examples are school returns to the (UK) Dept for Children, Families and Schools (DCFS), data collected on hospital admissions and waiting times etc, data collected by large supermarkets when shoppers use their loyalty cards and so forth. These are not representative samples, nor are they (in a major sense) populations. They are usually large data sets with many variables. Some are used to get indicators of performance and construct league tables. A knowledge and discussion of how these are constructed and of the positive and negative effects, the strengths and weaknesses should be part of the basic statistical education.

3 Levels of Understanding for Statistical Education

It is possible to identify three levels of understanding that are appropriate for every citizen and student. The first level comprises items that the student should know about. The second comprises items that the student should be able to identify and critically evaluate. The third comprises items that the student should be able to do.

A What a student should know about

This includes:

• the statistical process of solving problems, drawing inferences about populations from well designed experiments or well-chosen samples; that these inferences can be quantified using probabilities and some idea of what these probabilities mean;
• the sort of information collected in the National Census of Population and how this information is used; similarly for some other uses of official statistics;
• how statistics is used in industry – particularly in quality improvement processes;
• some current areas in which statisticians are actively working and the sort of problems they are solving; examples can be found in Significance and Chance magazines as well as the more serious press and other journals;
• the use of indicators to measure performance; their strengths and weaknesses;
• how businesses use large data sets;
• how statistics is used in their other academic courses;
• some technical terms that might be met in everyday reporting such as standard deviation and  confidence interval.

Note that largely these are areas where it would not be feasible for the students to do the statistics for themselves, but they should be able to discuss the issues on the basis of what they know.

B What a student should be able to identify and critically evaluate

This includes:

• newspaper and popular magazine accounts of an issue in which statistics was used;
• the use of statistics in other subjects;
• officially produced tables of data;
• graphs of data.

They would be expected to comment on such things as the nature of the sampling or experimental design, nature of questions on any questionnaire, whether the written description matched with the numerical or graphical presentation, were the important points made – were there any omissions, any misuses of statistics etc.

C What a student should be able to do

The appendix gives attainment targets for Handling Data from the 2006 National Curriculum for Mathematics, as interpreted by the Programme of Study for Key Stage 4. Even though these have now been replaced, there are aspects that teachers and students should find useful. In terms of update,  the 2008 Key Stage 3 programmes can be found here. For Key Stage 4 look here.

Activities in teaching should be focused on the major ideas of statistics, including using appropriate populations and representative samples, using different measurement scales, using probability as a measure of uncertainty, using randomness and variability, reducing bias in sampling and measuring, using inference to make decisions. Real data should be drawn from a number of contexts.

Appendix: A Programme of Study (QCA, December 2006)

Note: although the following programme of study has been replaced by a new one (2008, see the QCA web site), there are merits in using some of the old programme in teaching. 

Knowledge, skills and understanding

Using and applying handling data

1) Students should be taught to:
a.    carry out each of the four aspects of the handling data cycle to solve problems:
i.    specify the problem and plan: formulate questions in terms of the data needed, and consider what inferences can be drawn from the data; decide what data to collect (including sample size and data format) and what statistical analysis is needed)
ii.    collect data from a variety of suitable sources, including experiments and surveys, and primary and secondary sources
iii.    process and represent the data: turn the raw data into usable information that gives insight into the problem
iv.    interpret and discuss the data: answer the initial question by drawing conclusions from the data
b.    select the problem-solving strategies to use in statistical work, and monitor their effectiveness (these strategies should address the scale and manageability of the tasks, and should consider whether the mathematics and approach used are delivering the most appropriate solutions)

Communicating

c.    communicate mathematically, with emphasis on the use of an increasing range of diagrams and related explanatory text, on the selection of their mathematical presentation, explaining its purpose and approach, and on the use of symbols to convey statistical meaning 

Reasoning

d.    apply mathematical reasoning, explaining and justifying inferences and deductions, justifying arguments and solutions
e.    identify exceptional or unexpected cases when solving statistical problems
f.    explore connections in mathematics and look for relationships between variables when analysing data
g.    recognise the limitations of any assumptions and the effects that varying the assumptions could have on the conclusions drawn from data analysis.

Specifying the problem and planning

2) Students should be taught to:
a.    see that random processes are unpredictable
b.    identify key questions that can be addressed by statistical methods
c.    discuss how data relate to a problem; identify possible sources of bias and plan to minimise it
d.    identify which primary data they need to collect and in what format, including grouped data, considering appropriate equal class intervals; select and justify a sampling scheme and a method to investigate a population, including random and stratified sampling
e.    design an experiment or survey; decide what primary and secondary data to use.

Collecting data

3) Students should be taught to:
a.    collect data using various methods, including observation, controlled experiment, data logging, questionnaires and surveys
b.    gather data from secondary sources, including printed tables and lists from ICT-based sources
c.    design and use two-way tables for discrete and grouped data
d.    deal with practical problems such as non-response or missing data.

Processing and representing data

4) Students should be taught to:
a.    draw and produce, using paper and ICT, pie charts for categorical data, and diagrams for continuous data, including line graphs (time series), scatter graphs, frequency diagrams, stem-and-leaf diagrams, cumulative frequency tables and diagrams, box plots and histograms for grouped continuous data
b.    understand and use estimates or measures of probability from theoretical models, or from relative frequency
c.    list all outcomes for single events, and for two successive events, in a systematic way
d.    identify different mutually exclusive outcomes and know that the sum of the probabilities of all these outcomes is 1
e.    find the median, quartiles and interquartile range for large data sets and calculate the mean for large data sets with grouped data
f.    calculate an appropriate moving average
g.    know when to add or multiply two probabilities: if A and B are mutually exclusive, then the probability of A or B occurring is P(A) + P(B), whereas if A and B are independent events, the probability of A and B occurring is P(A) W P(B)
h.    use tree diagrams to represent outcomes of compound events, recognising when events are independent
i.    draw lines of best fit by eye, understanding what these represent
j.    use relevant statistical functions on a calculator or spreadsheet.

Interpreting and discussing results

5) Students should be taught to:
a.    relate summarised data to the initial questions
b.    interpret a wide range of graphs and diagrams and draw conclusions; identify seasonality and trends in time series
c.    look at data to find patterns and exceptions 
d.    compare distributions and make inferences, using shapes of distributions and measures of average and spread, including median and quartiles; understand frequency density
e.    consider and check results, and modify their approaches if necessary
f.    appreciate that correlation is a measure of the strength of the association between two variables; distinguish between positive, negative and zero correlation using lines of best fit; appreciate that zero correlation does not necessarily imply 'no relationship' but merely 'no linear relationship'
g.    use the vocabulary of probability to interpret results involving uncertainty and prediction [for example, 'there is some evidence from this sample that ...']
h.    compare experimental data and theoretical probabilities
i.    understand that if they repeat an experiment, they may - and usually will - get different outcomes, and that increasing sample size generally leads to better estimates of probability and population parameters.
j.    discuss implications of findings in the context of the problem
k.    interpret social statistics including index numbers (for example , the General Index of Retail Prices); time series (for example, population growth) and survey data (for example, the National Census)

Breadth of study

1) During the key stage, students should be taught the Knowledge, skills and understanding through:
a.    activities that ensure they become familiar with and confident using standard procedures for the range of calculations appropriate to this level of study
b.    solving familiar and unfamiliar problems in a range of numerical, algebraic and graphical contexts and in open-ended and closed form
c.    using standard notations for decimals, fractions, percentages, ratio and indices
d.    activities that show how algebra, as an extension of number using symbols, gives precise form to mathematical relationships and calculations
e.    activities in which they progress from using definitions and short chains of reasoning to understanding and formulating proofs in algebra and geometry
f.    a sequence of practical activities that address increasingly demanding statistical problems in which they draw inferences from data and consider the uses of statistics in society
g.    choosing appropriate ICT tools and using these to solve numerical and graphical problems, to represent and manipulate geometrical configurations and to present and analyse data.
h.    Consider how in real life to make informed decisions, and recognise the difference between meaningful and misleading representations of data