Brief Description

This unit introduces the capture-recapture method of estimating population sizes and uses this to look at some of the problems of estimating and the properties of estimators.

Design Time: About 4 hours.

Aims and Objectives

On completion of this unit pupils should be able to estimate populations using the capture-recapture method which is used in biology. They will have practised this through simple examples using proportion and harder examples using a formula. They should be more aware of the effect of sample size and various practical problems on the accuracy of estimates. They should appreciate that any estimate is liable to be inaccurate and that there is variability amongst estimates.

Prerequisites

Pupils will be assumed to have some basic knowledge of proportion, to be able to manipulate simple fractions and cancel common factors, and substitute into and evaluate a simple fractional formula. They should also be able to work out simple means.

Equipment and Planning

Several hundred small beads or cubes of different colours are needed in Section C. Sampling bottles could be used if available. Section D1 needs prior planning. Pupils will work individually except in Section C where small groups are required.

Detailed Notes

Section A

It is hoped that a class discussion will show that the problem we are dealing with is real and will help children to see some of the difficulties involved.

Some of the topics can be followed up by reference to articles on 'The Plight of Whales' in Statistics: A Guide to the Unknown (ed. J. Tanur) and 'Estimating the size of Wildlife Populations' in Exploring Data (ed. F. Mosteller, et al.).

Methods other than capture-recapture for the whale, cod or herring problem involve examining the rate of catching over the years or an analysis of the age of the fish caught. This can be deduced from the number of dark bands in the growth rings of the fishes' scales.

The emphasis on estimating, not counting, is essential. Counting is impossible and estimates will vary.

Section B

This section spells out the basic principles of the capture-recapture method. The importance of approximation cannot be overstated. There are many assumptions made in the use of proportion in the second catch and some are spelled out in Section D2. It would be useful in B1 to give specific figures for the second catch (e.g. a catch of 40 with 11 marked red) and see the implications of trying to put ¹¹/₄₀ = ¹/₄

B1
Waiting for one day was necessary to allow the fish to mix (an important point which should be stressed here, though the point of waiting is brought up in later questions). This makes it reasonable to assume that each fish has an equally likely chance of being caught in the second sample.

The format of Table 1 is used to help pupils see the pattern of the numbers. Initially we consider a pond where the numbers of fish is known, since this makes the process easier to visualize and enables the basic principles to be laid down. There are two ways of looking at the figures in this table to get an estimating procedure. One, based on the idea of the proportion of marked fish in the pool and in the second sample, is spelled out in the pupil notes between g and h. The second is to say that the second sample contained, ⁴⁰/₁₀₀ of the fish in the pond, so that we expect it to contain, ⁴⁰/₁₀₀ of the marked fish, giving the answer 10 as before.

7. There are both practical and statistical problems here. The practical ones will be dealt with in D2. The statistical ones are:
   1. It may be numerically impossible to get exactly the right answer, i.e. in a second sample of 15 where the proportion is ¹/₄:
      ^x/₁₅ = ¹/₄ has no solution with x as a whole number.
   2. The inherent variability in the number of marked fish caught the second time leads to a number of different estimates on different occasions.

B2
Here we move from a pond containing a known number of fish to one where the population is not known.

The same basic principles apply as in B1, and the number patterns may well be seen from Table 2 in many forms, e.g. (in algebraic terms) if the table is:

Then some will see ^c/_N = ^r/_s.
Others will see or ^s/_N = ^r/_c
or Nr = cs,
or ^c/_r = ^N/_s,
or ^N/_c = ^s/_r

It is this last formula that is followed up in the text.

Figure 1 illustrates the same figures as Table 2 but in a less abstract way and this may be used to help the weaker pupils who may find this section slightly difficult. Blank Table 5 on page R1 is used for the worked example in a to d. Further copies of this blank table can be drawn up to help with the remaining examples. All examples from e to j are optional for reinforcement of the pattern. Weaker pupils might copy the worked example as a guide to the method.

B3
This section brings out the formula more explicitly. Pupils with a background in more formal algebra may like to let:

c = first sample size (captured)
s = second sample size
r = number of marked fish in the second sample (recaptured)
N = number of fish in the pond

and get the formula N = ^cs/_r.

One way of getting this is shown formally in the text. Other methods are transformations of the formulae described in B2. It is better if children see the pattern from the numbers rather than learn a formula.

More difficult numbers are also introduced to emphasize the fact that the answers are only estimates and it is sensible to give estimates to whole numbers only.

6. The occurrence of no marked fish in the second example gives an estimate of an infinitely large number of fish when using the formula, or intuitively no estimate at all (except that it is more than the sum of the two samples). This might happen if the first sample were small compared with the overall population. When testing this work in schools, one girl suggested that it would happen if the marked fish were hiding. This led to an interesting discussion on whether fish were equally likely to be caught a second time. This is followed up in Sections D and E.
7. This would happen if the first sample netted all or nearly all of the fish in the pond. The estimate would be 'about the number of fish captured at first'.

Section C

There are three main points behind the simulation.

1. To show the effect of sample sizes on the reliability of estimates.
2. To emphasize the many (sometimes unrealistic) assumptions made in the capture-recapture process. This point is taken up again in the next section.
3. To show that there can be a considerable range of estimates, and that a good estimate should have its distribution centring on the true value with as small a spread as possible.

   Working in small groups, of say four, is essential here to avoid an important experiment becoming tedious.

   It is suggested that each group has the same number (around 100) of beads in its container for C1 to C4, and thus groups might 'compete' to see who can achieve the closest estimate.

There are many ways of arranging this work. C1 and C2 keep the same capture size but change the recapture size. C3 and C4 also keep a constant (but higher) capture size and change the recapture size. It would be useful if all groups did two of C1 to C4 and that each of C1 to C4 was done by at least one group. Groups doing C1 and C2 or C3 and C4 would see the effect of changing the recapture size. Groups doing C1 and C3 or C2 and C4 would see the effect of changing the capture size.

It is useful to collect and display the class results as a line chart such as in Figure T1. In this way the effect on the estimates can clearly be seen.

Figure T1 - Estimates based on a capture size of 20, a recapture size of 40

The variation in the size of r is important to show the considerable degree of variation in estimates, especially for c = 20.

Some groups might like to draw one or two line charts to show the distribution of their estimates around the true value.

Section D

D1
The practicality of this example depends on the nature of the school. It does pose organizational difficulties, but if it can be attempted it emphasizes very well many assumptions made and problems faced in actually using the capture-recapture method.

Two ways are suggested for marking pupils.

1. The sampler gives the sampled pupils a coloured counter (if more than one group does the sampling, each could use a different colour). The recapture sampler will go out the second day (or later the same day) to ask how many of his sample have his coloured counter.
2. On the first day the sampler asks the sampled pupil his name and form. On the second day another sampler does exactly the same. Those on both lists are the recaptured ones.

It is important that different pupils go out on two successive days and that some randomization of interviewing is attempted. This may be done by randomizing the position to which the interviewers go each day. Pupils should be warned not to interview only those of the same sex, same age group, etc.

Sample sizes should be a reasonable proportion of the total population being sampled in order to get a reasonable number of 'recaptures'.

If the biology department has a school pond or does a capture-recapture experiment elsewhere, this could take the place of D1.

D2
This section is designed to get the children thinking about the practical difficulties involved, and the ways that the simple mathematical model used does not reflect reality. Answers to some of these questions are qualitative - along the lines of: 'This will make the estimates a little less reliable and they will tend to be underestimates.'

1. Other assumptions made, but not specifically mentioned, are that we are able to capture a reasonable proportion of the whole population at the first attempt, and that this is a random sample.

Points which affect the assumptions include the following:

Birth, death, immigration and emigration processes will change the population.

Any shoaling effect of the fish would upset this assumption.

There may be many reasons for the third assumption to be untrue. Marked fish may have been harmed in the marking. The fact that they have been caught once may show that they are more likely to be caught anyway or that they will be more wary next time. If the sampling is done at the same place each time then, unless there is complete mixing, those fish living in inaccessible places won't be caught; and so on. Other problems may be connected with the actual marking and finding the marks on the second sample.

Section E

This section can be treated as an exercise to see whether or not the ideas of the previous section have been assimilated.

E1
Reliability is poor because of the small number of marked fish in the second sample. A more reliable estimate would be obtained by taking either a larger first sample or a larger second sample.

E2
Setting traps in the same place each time means you are more likely to catch the same squirrels and not likely to catch squirrels in a different part of the wood, so the original estimate is an underestimate.

E3
Over two years there would be deaths and migration of marked squirrels so that at the second sampling the true number of marked squirrels is less than those originally marked. This means that we overestimate the number of squirrels in the wood. The effect of the hunters is in the same direction.

E4
A number of marked whales in the second sample would be counted as unmarked. This leads to an overestimate of the population. With the growth of conservation movements such as 'Green Peace' the problem of assessing the numbers of whales is likely to become a more emotive issue.

Answers

B1	a	The number of fish in the pond and the number of fish in the second sample respectively.
	c	1/5
	e	1/5
	f	No
	g	No
	h	See detailed notes.
B2	a	24 fish
	c	(12) 6 (2)
	d	30 fish
	e	21 fish
	f	60 fish
	g	24 fish
	h	90 fish
	i	100 fish
	j	120 fish
B4	a	21 fish
	b	25 fish
	c	30.3 = 30 fish
	d	37.5 = 38 fish
	e	16.8 = 17 fish
	f	See detailed notes.
	g	See detailed notes.
E1	a	2500 trout
E2	a	18 squirrels
	c	See detailed notes.
E3	a	50 squirrels
	b	See detailed notes.
	c	See detailed notes.
E4	a	See detailed notes.

Page R1

	In the pond	In 2nd sample
Marked fish
Fish altogether

Table 4

Table 5

Table 6

Table 7

Test Questions

1. In a pond a sample of 10 fish is caught and marked, then returned to the water. Next day a second sample of 12 is taken, and four fish are found to be marked. Estimate how many fish are in the pond. Give a reason why your estimate might not be accurate.
2.  
   1. Twelve beads are taken from a bag, marked and then returned. A second sample is taken, after the beads have been mixed, and of 20 beads four are marked. Estimate the number of beads in the bag.
   2. Two bags each contain the same number of beads. Ann takes a first sample of 20 from her bag, marks them and returns them to the bag. Brian does the same with a first sample of 40 from his bag. They both take second samples of the same size and estimate the number of beads. Whose estimate is likely to be more accurate?
3. From a bag containing some beads a first sample of 30 is taken, marked and returned. Ten people come and take a second sample of size 20. The number of marked beads in the second samples were:
   5, 6, 6, 8, 4, 5, 7, 6, 1, 3
   1. For each sample estimate the number of beads in the bag.
   2. What are the largest and smallest estimates.
   3. Use all these samples to give your estimate. Explain how you worked it out.
4. Along a 400-metre stretch of river 100 fish are netted, marked and returned to the water. A week later a second sample of 150 is caught, and 20 are found to be marked. Estimate how many fish are in the stretch of river. Give any reasons why you think your estimate might be unreliable.
5. On the 1st April 1979 a trapper caught 50 rabbits in sand dunes. He put rings round their legs to mark them. On 1st April 1980 he caught 60 rabbits in the same place and found that four of them had rings on. Use the capture-recapture method to estimate the number of rabbits in the dunes on 1st April 1980. Say, with reason, whether you think this estimate is likely to be too high or too low.
6. The following table shows the results of using the capture-recapture method to estimate the number of beads in a bag.
   

   	1st sample	2nd sample	No. of marked beads in 2nd sample
   Alan	40	20	12
   Claire	10	5	0
   David	15	15	2
   Elaine	60	40	27

   1. Which estimate is likely to be nearest the correct answer? Why?
   2. Which estimate is likely to be furthest from the correct answer? Why?

Answers

1		30 fish. Small number of marked fish in second sample, etc.
2	a	60 Beads
	b	Brian's
3	a	120, 100, 100, 75, 150, 120, 851/2 (86), 100, 600, 200
	b	Largest 600 - Smallest 75
	c	The mean of the other estimates which is 165. Also acceptable is the median 110.
4		750 fish - marked fish go out of the stretch of water, others enter.
5		750 rabbits - probably too high because of death and migration.
6	a	Elaine (large first sample and larger second sample).
	b	Claire (small first sample and second sample and no marked beads in the second sample).

Connections with Other Published Units from the Project

Other units at the Same Level (Level 3)

Car Careers
Cutting it Fine
Multiplying People
Phoney Figures
Pupil Poll

Units at Other Levels In the Same or Allied Areas of the Curriculum

Level 1

Shaking a Six
Practice makes Perfect
If at first ...

Level 2

Seeing is Believing
Getting it Right

Level 4

Smoking and Health

This unit is particularly relevant to: Science, Mathematics.

Interconnections between Concepts and Techniques Used In these Units

These are detailed in the following table. The code numbers in the left-hand column refer to the item spelled out in more detail in Chapter 5 of Teaching Statistics, 11-16.

An item mentioned under Statistical Prerequisites needs to be covered before this unit is taught. Units which introduce this idea or technique are listed alongside.

An item mentioned under Idea or Technique Used is not specifically introduced or necessarily pointed out as such in the unit. There may be one or more specific examples of a more general concept. No previous experience is necessary with these items before teaching the unit, but more practice can be obtained before or afterwards by using the other units listed in the two columns alongside.

An item mentioned under Idea or Technique Introduced occurs specifically in the unit and, if a technique, there will be specific detailed instruction for carrying it out. Further practice and reinforcement can be carried out by using the other units listed alongside.

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