The inherent variability of biological material creates a mutual attraction between biology and statistics, an attraction which can lead to a deeper understanding of the two subjects. For a variety of reasons many teachers of biology are ill at ease with statistics; however the growing acceptance of statistics at school both as a subject in its own right and as an element of subjects such as biology is an encouraging sign for the future. Experimental work is recognised to be an essential part of the teaching of biology and some of this practical work could be of help in overcoming the quite formidable barriers which many pupils encounter in their struggles with statistics.

**A Classroom Pond**

Work with pond life can play an important part in the teaching of such
fundamental biological issues as the distribution of species, food chains,
and population dynamics. Our experience suggests that a variety of
pupils find this work interesting and meaningful. The difficulties
of working in the field can be eliminated by bringing pond life to the
classroom, and this makes the necessary practical work cheap and easy to
set up. From a statistical point of view, pond life can be used to
illustrate a wide range of concepts and methods.

Plastic sweet jars, which are often obtainable free from confectioners,
make ideal containers for small fresh water pond animals. The animals
which we have found most suitable are those from small temporary ponds
and ditches; these habitats usually provide a variety of abundant invertebrates
in densities far above those in the clean looking larger ponds. By
using a plankton net, as shown in *Figure 1*, one can obtain a concentrated
stock from which the pupils' ponds can be made. If no local pond
is available then the live Daphnia sold in pet shops for approximately
5p a bag will give a reasonable variety of specimens for one sweet jar
pond.

In order to learn about the distribution of species or the size of
a population one estimates by sampling. The simplest way of sampling
volumes from the sweet jar pond is to use a pipette made from a wide bore,
say 5 mm diameter, glass tube approximately 300 mm long, attached to a
disposable syringe by a small piece of plastic tubing. A range of
syringe sizes from 1 to 20 ml is suggested. *Figure 2* shows
this apparatus and the method of dividing the samples into small parts
so that the moving animals are restricted and are easier to count.

The apparatus is available from most school suppliers. The glass
rods and the petrie dishes can be obtained from the school science laboratory
and Health Centres may well help with syringes without needles.

If the identification of the small vertebrates is felt to be a problem, the work could be in terms of total number of animals in a sample. Two suggested books are Clegg (1956) and Macan (1959)

**An Example**

The difficulty of counting the total number of animals in the sweet
jar pond and a possible bias due to clustering is obvious so that "stir
the pond and count the number in a small amount of water" is the sort of
response that occurs.

For the younger age group this sort of work will provide the data for a graphical summary and a motivation for calculating the mean value of a set of numbers. If the pupils are divided into groups, a comparison of the results from different groups may demonstrate some of the facts of life about variability. Qualitative differences in the level of interest and the degree of care taken in obtaining the results will emerge. The time taken to carry out the experiment and the estimates of the size of the population will vary from group to group. Reasons for any observed major differences between groups could provide opportunities for an enriching experience. If an undisclosed estimate of the number of animals from a previous practical, preferably a large scale one, is available, the pupils would be interested in the closeness of their answers to the 'master' estimate. For the older age group this work opens up a number of avenues, and we illustrate one important avenue through some data obtained by a group of teachers in a series of experiments at Southampton University. The teachers were enthusiastic and of mixed ability in biology as well as statistics.

An appreciation of the Poisson distribution and an examination of a
set of results from the point of view of consistency with some assumed
distribution are important theoretical as well as practical issues.
An understanding of the conditions that give rise to a Poisson distribution
will result in the use of this distribution for pond life data. A
chi-squared (c^{2)} test
is a commonly used one for testing the goodness of the fit of an assumed
distribution. Details about the Poisson distribution and chi-squared
test are given in many books on statistics and we suggest Campbell (1974),
Hayslett (1971), and Hodge and Seed (1977).

No. of Daphnia | Observed frequency (O) | Expected frequency (E) | (0 - E)^{2}/E |

<=6 | 9 | 8.40 | 0.043 |

7 | 6 | 5.24 | 0.110 |

8 | 4 | 6.20 | 0.781 |

9 | 7 | 6.51 | 0.037 |

10 | 10 | 6.16 | 2.394 |

11 | 5 | 5.30 | 0.017 |

>=12 | 9 | 12.19 | 0.835 |

Total | 50 | 50.00 | 4.217 |

For the younger age group the procedure of mixing the pond, taking the
mean of the number of observations and multiplying it by the volume of
the water (expressed in suitable units) in the jar will seem to be the
right thing to do. For some older pupils interested in statistics
as a subject, the action of mixing *N* (unknown) *Daphnia* in
a volume *V* of water and noting the number of *Daphnia* in,
say, volume *k* provides an interesting experience of a Poisson process.
It is certainly a clearer realisation of a Poisson process than counting
the number of passing cars per unit time. The pupils will readily
appreciate that mixing the *Daphnia* is a physical analogue of "events
occurring at random in a continuous medium". The average number of
*Daphnia* per volume *k* is *m* = *kN/V*, so that we
can predict that the proportion of times that exactly *r Daphnia*
will be observed is the Poisson probability

P(*r*) = *m ^{r}e^{-m}/r!*,

The mean value of the observed results will estimate *m* so that
the use of the factor *V/k* will estimate *N*.

**Pond Potential**

Work with the sweet jar ponds can illustrate a variety of biological
and statistical ideas. A study of some of the factors that influence
the animals in a pond can be made by setting up suitable experiments.
For example a study of acidity can be made by adjusting the *pH* value
of a sweet jar pond through the addition of buffer solution tablets.
Pond life may be influenced by the various substances of the buffer tablets,
so that any observed effects may not be entirely due to acidity.
We suggest a range of pH values from 4 to 8. At least seven days are needed
for the animal population to stabilise. Temperature, and light are
two other factors that can be studied.

As an aid to understanding statistical ideas, this work has a very rich potential, especially for the older age group of pupils. Some of the possibilities that come readily to mind are:

1 . A consideration of sample size. Is it better to take 2ml or 10ml samples? Are two samples of 5 ml equivalent to a single sample of 10 ml?

2. Appreciating sums of random variables. Many teachers of statistics will agree that pupils often find this a difficult concept. We make some further comments about sums of random variables later.

3. A manifestation of central limit theorem which predicts that under a fairly wide range of circumstances, summing of random variables leads to a normal distribution.

4. Formulating and testing meaningful hypotheses. Here co-operation between teachers of biology and statistics would be most helpful.

5. Generating data for some of the standard statistical methods. We have already discussed the fitting of a Poisson distribution. There are many other possibilities. For example, confidence limits for the population size of a particular sweet jar pond; an examination of the differences between the populations of two jars; the changes in the population over time or due to factors such as pH.

We now return to the difficult concept of sums of random variables. An experiment in which the results of, say, 2 ml syringes are used to create data about 4 ml syringes is a physical demonstration of the sum of two random variables. For example, given four results 3, 4, 4, 0 with 2 ml syringes, the pupils would readily appreciate that these generate two observations 7(3 + 4) and 4(4 + 0) for the 4 ml syringes. Two groups of pupils, one group working with 4 ml and the other group with 2 ml syringes could summarise their results in the form of histograms and hopefully these histograms would show that the sum of two 2 ml results yields the same sort of histogram as the 4 ml results. In other words, the sum of two Poisson random variables is a Poisson variable. This experiment should be supplemented by an experiment where the action of summing changes the nature of the random variable. For example, the sum of two rectangular variables is not a rectangular variable as may be illustrated by throwing a pair of dice, or by using random digits.

**Acknowledgements**

This paper has developed from the work carried out by P. S. Parsons
as a B. P. Research Fellow at the University of Southampton.

Financial assistance from the *School Mathematics Project* made
it possible for a group of teachers to come to a series of meetings at
Southampton University. We have enjoyed working with this group.

We thank a referee for making a number of helpful suggestions.

**References**

Campbell, R. C. (1974). *Statistics for biologists.*
C.U.P.

Clegg, J. (1956). *Observers book of pond life*. Frederick
Warne.

Hayslett, H. T. (1971). *Statistics made simple*.
W. H. Allen.

Hodge, S. E. and Seed, M. L. (1977). *Statistics and Probability.*
Blackie Chambers.

Macan, T. T. (1 959). *A guide to freshwater invertebrate animals.*
Longmans.

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