between the effect of the hand on the timing and the fact
that the first effort gives practice so the second may be
quicker because it is second.
TEACHING STATISTICS 8. MAKING YOUR A-LEVEL STATISTICS TEACHING PRACTICAL Peter Holmes Centre for Statistical Education University of Sheffield
8.1 Introduction
Over the past years there has been an increasing emphasis on doing practical work as part of A-Level Statistics and greater pressure placed on teachers to teach in this way. Some syllabuses require all candidates to do project work, and these projects must include practical work. It is not acceptable for them to do just theoretical projects. Why is this so and what can we do to make our teaching more practical?
Teaching statistics only as a set of techniques or as mathematical theory is rather like teaching only the skeleton in a human biology course. The techniques are only the bones of statistics. How can we make these bones live? One way is through practical and project work. The Joint Matriculation Board, in its notes for guidance to teachers on A-Level Statistics, gives five main reasons why projects are a good idea.
2. Projects give more motivation. Particularly this is true if students are allowed to choose the subject of the project - choosing something that they really want to know about. One interesting project I saw was actually done by primary school pupils as part of the Annual School Statistics Prize. The pupils at Moore County Primary School lived in a small village where there was no gas. They carried out a survey of people in the village to find out their views, how much they would use gas if it were available, how much they would be prepared to pay to have gas brought to the village and so on. Being a Primary School project it did not use difficult techniques, but it was a careful analysis. They submitted their report to British Gas and I heard two years later that the decision had been made to put the village on the gas network
3. Projects give a greater feel for real data; its accuracy or otherwise; variability; reliability of conclusions; measurability. It is easy to carry out, say, a confidence interval calculation on a collection of data without considering where the data come from. Yet this can be important in how much confidence you really put in the results. For example, giving a confidence interval of a proportion based on 55 people out of 100 saying they would vote for a particular party is the same calculation as having 55 out of 100 coins weighing more than a particular weight. Yet when you collect the data from people you may be aware that borderline cases are shading their answer to what they think you want. The data are inherently less accurate.
4. Projects emphasise the application of statistics and its usefulness. This will become clear to a student from his/her own project, and he/she will also benefit from the different projects being done by fellow students. In the projects they are forced to draw practical conclusions
5. Projects show that statistics is not solely mathematics. In particular the choice of the sample, the design of the experiment or the questionnaire, the practical summary at the end, all require skills that are more than mathematical.
Anderson and Loynes (1987) give a much more detailed analysis of the skills and abilities that are developed by doing practical work. Even if your particular syllabus does not require you to do practical or project work as part of the assessment it is still worthwhile doing it as part of the teaching because of the spin-off for understanding and insight that students gain.
8.2 Practical or Project Work
There are differences between practicals and projects. It may be that you do not have time to have students doing a full-scale project on their own, but you could fit in different practical sessions into the lessons. A project may take many hours work (perhaps 40 or more); a practical is typically short and, with analysis, may take up to 1.5 hours. Other differences are shown in the table below.
| Practical | Project |
| Introduces or reinforces theory | Links topics |
| Has clear cut objectives | Is investigative |
| Teacher defines problem | Student defines problem |
| Teacher decides model | Student chooses model |
| Teacher decides techniques | Student decides techniques |
| Teacher leads discussion | Student draws own inferences |
From this it is easy to see that practicals are much more under teacher control and can be geared more easily to specific topics in the syllabus. It is easy to build up a library of practicals that can link to different topics.
8.3 Different types of practical
a. Active or passive
In an active practical the student is directly involved. In a passive practical the student watches a demonstration. There is room for both types, and for things in between, but generally the student gains more from active involvement. Passive demonstrations are best kept for those practicals which are too difficult, expensive etc. for the students to do themselves. Video recordings of people doing statistics as part of their work would come in this category.
b. Closed or open
A closed practical is linked to specific results; an open practical allows more investigation. For example a practical which says 'Throw 5 coins. Count the number of Heads. Repeat 256 times. What is the distribution of the number of Heads? Can you explain the distribution?' is a closed practical leading to B(5, 0.5). A practical which says 'What happens when you throw coins?' is open and could lead to relative frequency definition of probability, to the binomial distribution if the number of Heads in a fixed sample size is considered, the negative binomial if the number of throws to get, say, 5 Heads is considered etc.
c. Design of experiments
Practicals can be used to help students see how to make the
best use of data by proper design considerations. For example,
suppose we want to use a reaction ruler to decide whether someone
is quicker to react with their dominant hand or their non-dominant
hand. (A reaction ruler is like an ordinary ruler except the
graduations give the reaction times in hundredths of a second.
They are used by dropping through a person's hand and seeing how
long it takes for them to react to catch it.)
| If everyone first uses the
non-dominant hand and then the dominant hand (in that
order) then there is confounding between the effect of the hand on the timing and the fact that the first effort gives practice so the second may be quicker because it is second. |
|
d. Introduce or reinforce theory
For example the relative frequency definition of probability considers an experiment carried out under the same conditions. What does this mean in terms of, say, a coin being tossed. By trying to reproduce the same conditions (perhaps with a machine) can we increase the probability of a Head to, say, 0.8? And how different may these conditions be? For example what happens to the relative frequency of the results of premier league football matches as you consider more and more games? In fact the proportion of home wins settles down to give us an estimation of the probability of a home win. These games are carried out under far from identical conditions.
e. Illustrate standard distributions
In our book Practical Statistics, Mary Rouncefield and I developed a practical to introduce the binomial distribution with p = 0.5. It consists of 7 statements; the pupils have to say whether they are true or false. The statements were chosen, and tested, to make sure that it was highly unlikely that the students would know the answer. One statement, for example, is 'A coho is a fish - True or False'. Collecting the results and finding how many students get 0, 1, 2, 3 etc. correct gives a motivation for looking at the binomial distribution, in particular B(7,0.5).
f. Clarify concepts
Crucial to developing a feel for the appropriateness of the standard statistical distributions as models is a consideration of the underlying assumptions. If we are to use these distributions as models when the underlying assumptions may not quite be true we need to have a feeling for how robust they are against deviations from these strict assumptions. One practical that I have used in this connection is to look at the distribution of the number of s's in lines of text in a long piece of prose. The argument goes that each line is (almost) the same length. Because type is proportionally spaced there will not be exactly the same number of characters in each line, but the number is fairly large. The probability that any particular character is an 's' is fairly small. Whether this probability is independent of the other letters that occur near it is highly doubtful. So if we see whether the number of s's in lines of text follows a Poisson distribution we come with a certain amount of scepticism.
Below are the actual and theoretical results for a particular piece of text from Teaching Statistics.
| No of s's per line (x) | No of lines (f) | Expected frequency | c 2 |
| 0 | 1 | 1.97 | |
| 1 | 6 | 8.39 | 1.090 |
| 2 | 19 | 17.90 | 0.068 |
| 3 | 28 | 25.44 | 0.258 |
| 4 | 25 | 27.12 | 0.166 |
| 5 | 24 | 23.13 | 0.0323 |
| 6 | 19 | 16.44 | 0.399 |
| 7 | 11 | 10.02 | 0.096 |
| 8 | 5 | 5.34 | 0.096 |
| 9 | 2 | 2.53 | |
| Total | 2.204 |
The sample mean is 4.26 and the expected values are calculated
from the Poisson distribution with this mean. The diagram shows
that they are fairly close and the calculated c 2 value is
fairly small. Maybe this distribution is fairly robust against
the 'independence' assumption. More exploration is needed!
Observed (clear) and expected (shaded)
frequency of s's
g. Practicals and computers
There are many ways in which computers can be
linked with practicals and be used as a tool for practical
investigation. Simulation of experiments is one obvious example.
Here is another where the spreadsheet has been set up to find the
binomial probabilites for different values of n and p
. By changing the values of p the probabilities change and
so do the graphs. Exploring in this way can give a great deal of
insight into how the binomial distribution depends on n and
p. The following table illustrates, for fixed p=0.3, the
dependency of the binomial distribution on n.
| Binomial Distribution | |||||
| n = | 5 | 10 | 20 | 40 | |
| p = | 0.3 | ||||
| r = | 0 | 0.16807 | 0.028248 | 0.000798 | 0.000001 |
| 1 | 0.36015 | 0.121061 | 0.006839 | 0.000011 | |
| 2 | 0.3087 | 0.233474 | 0.027846 | 0.000091 | |
| 3 | 0.1323 | 0.266828 | 0.071604 | 0.000495 | |
| 4 | 0.02835 | 0.200121 | 0.130421 | 0.001963 | |
| 5 | 0.00243 | 0.102919 | 0.178863 | 0.006057 | |
| 6 | 0 | 0.036757 | 0.191639 | 0.015143 | |
| 7 | 0 | 0.009002 | 0.164262 | 0.031522 | |
The following graphs were embedded in the spreadsheet.
Binomial probabilities for n=5, p=0.3
Binomial probabilities for n=20, p=0.3
Changing p immediately changes the calculations and the
graphs. Other examples such as illustrating the Central Limit
Theorem can easily be devised.
Practical and project work can bring your statistics teaching to life. They are well worth while the extra effort required to incorporate them into courses.
8.4 References
C W Anderson & R M Loynes (1987). The Teaching of Practical Statistics. Wiley, Chichester.
M Rouncefield & P Holmes (1989) Practical Statistics. Macmillan, London.