Practical experimentation in biology can throw up problems of data collection, presentation and analysis.  The author describes a simple experiment and shows how the problems raised were tackled with some 13-year old pupils.
 

Introduction
It is increasingly accepted that the use of real data, and the experience gained in planning and conducting statistical experiments, are essential components in the teaching of statistics from the earliest stages.  Whilst attending a series of meetings at the Mathematics Department of Southampton University, I attempted this fairly ambitious and long-term project with these considerations in mind using a 3rd form group at my school.  It was ambitious both in its length and in the relative complexity of some of the material, but at the end I felt that it was very worthwhile.  It had enabled me to teach in a very practical way; too often at this level, descriptive statistics can be boring and rather pointless.  This time, the class had been continually involved in the search for a satisfying answer to the question posed. I hope this article will give other teachers ideas for projects for pupils in this age range.

The Project
A fairly simple problem from Biology was taken as the basis of the project, so making the interdisciplinary nature of the applications of statistics apparent at an early stage.  The problem was to try to form an opinion on whether populations of Daphnia and Bristlebacks (Thysanura, small water creatures), cluster or whether they position themselves substantially at random.

The pupils must be involved in the planning of the experiment from the earliest stages.  Much can usefully be taught in attempting (and probably failing), to cover all future eventualities.  The first problem is to record the positions of the Bristlebacks at a particular instant for later analysis.  There is really only one technique here photography, but the manner of taking the photographs immediately poses many other problems, which with guidance the pupils discover for themselves.  For example:

(i) Will the depth of liquid in which they are photographed matter?
(ii) Will the position and strength of lighting matter?
(iii) How long should they be left after any physical disturbance of the liquid before the photograph is taken?
(iv) What is the minimum population size that should be used?
(v) What film, speed and background will give the best results, and how many photographs should be taken and analysed?

These problems and others (the list is not exhaustive) will be shared and discussed by teacher and pupils, and overcoming them or reaching compromises is a very useful exercise. I am deliberately not giving precise details of our answers to these problems here, as I feel it is of value, in a project of this sort, for the class to see the teacher facing up to the realities of experimentation. I have simply outlined the nature of the problems that occur.

Obtaining the Data
Having obtained the photographs, the next stage is to analyse the positions thus recorded, and then to judge these for clustering.  By careful questioning along the lines: 'How many think there are more in the bottom left corner than elsewhere?' and 'How could I record that?' the use of a grid overlay on the photographs dividing the area into cells comes out naturally from the discussion.  At this stage I then gave each pupil photocopies of the photographs for them to construct their own grid system on. I suggested a system giving an average of about one Bristleback per cell, but variations were permitted for those who wanted to test my suggestion of seeing the effect of changing the value of this parameter.  Frequency tables of the number of Bristlebacks per cell were then constructed, but of course, new problems immediately arose-what to do with those inconsiderate Bristlebacks that straddled two or more cells, for example?  This again provides useful teaching material in respect of how to cope with these unforeseen problems.  The answer here must be to make a firm decision such as 'count it in the cell where the majority of its body area lies', or 'always count it in the lower cell if it is exactly halfway', before starting the analysis.  Problems of this type will inevitably occur when real data is being collected, and in my opinion the recognition and the solution of them is important, but frequently missed in the teaching of statistics at this level.

Looking at Results
The pupils now have data from which they can present frequency tables and graphs, and perhaps calculate a mean number of Bristlebacks per cell.  Such a set of results is shown in the figure.  But now, how much further can we go?  A chi-squared test to a Poisson distribution is obviously inappropriate here, as a knowledge of the theoretical expected distribution requires A-level mathematical techniques, but in response to the probe 'How do we judge if they are definitely not randomly positioned?' The answer 'Compare against a known random pattern' is quite easy to obtain from the pupils.

This leads to another interesting section of the project - the generation of an accepted random pattern of the same population size as the Bristlebacks.  All the pupils' suggestions for doing this should be tried if time permits, from a pin with closed eyes, to using dice to generate a pair of random co-ordinates for a similar cell system as was used before.  Both of these have limitations; for instance, dice really limit the simulation to 36 cells, unless 10 sided (or more) dice are used, and for direct comparison with the live data more than 100 cells are likely to be needed, especially for the more numerous Daphnia.

I took the opportunity to introduce random number tables at this stage.  It seems to me that the earlier these are introduced and accepted the better.  Here, their use has the advantage that there is a definite aim in mind; they are a means to an end, and not an end in themselves.  The pupils checked with frequency tables of single digits that they behaved as random digits should behave.  Working systematically from any starting point in the table (see for example, SMP advanced tables, p. 49), the frequencies of each digit occurring in a total of say 100 digits is recorded.  If the table is truly' random', then one expects each of the 10 digits to come up with about the same frequency.  This will give an approximately flat histogram no matter which starting point is chosen.  Pairs of random digits are then used to produce pairs of co-ordinates (x, y) which are then used to simulate the random pattern of Bristleback positions.  These patterns were then analysed in the same way as the experimental data, and the frequency graphs superimposed for an easier comparison to be made.  The pupils are then in a position to judge (albeit still qualitatively) the answer to the original problem-are the Bristlebacks clustered?
 
 

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