Investigation On How The Sample Size Affects Accuracy

Published Date: 02 Nov 2017

Lincoln index is a means of estimating population. It is also called the Capture - Mark – Release – Recapture method. Population is calculated using the following equation:

Total Population =Number caught x Number re-caught

Number marked in the re-catch

Or: N = (n1 X n2) / n3 Where N =Population Estimate

n1 = Number caught initially

n2 = Number re-caught

n3 = Number marked in the re-catch.

Research Question:

How does the sample size affect the accuracy?

Hypothesis:

Sample size in the Lincoln Index, is the size or the number taken in the species, which is marked and released backed into the population. The higher the sample size taken, the better chances of getting a more accurate population estimate. As the sample size reflects the size of the population, when it increases, the number re-caught are more likely to represent the actual population size more accurately than when done with a smaller sample size.

Materials Required:

1 A4 sized sheet of white paper

Scissors

Pen/ Marker

Paper to collect data

Procedure:

Cut up an A4 piece of paper into pieces around 2cm x 2cm.

Put them in a pile. Remove 10 pieces and mark them (n1).

Release the pieces of paper back into the pile. Mix up the pile.

With your eyes shut pick out 10 pieces of paper. Record the amount of re-catch (n2).

Record the number of marked pieces of paper (n3)

Repeat stages 4 and 5 five more times.

Take out 10 un-marked pieces mark them and release them into the pile.

Repeat stages 4 to 6 but with 20 pieces of paper.

Repeat until you have looked at least 5 sample sizes.

RAW DATA:

Table 1: Table showing data collected for Sample Size 10.

SAMPLE SIZE:

10

Number caught initially (n1)

Number re- caught (n2)

Number marked in the re-catch (n3)

10

10

1

10

10

1

10

10

1

10

10

2

10

10

1

Table 2: Table showing data collected for Sample Size 20.

Table 3: Table showing data collected for Sample Size 30.

SAMPLE SIZE:

20

Number caught initially (n1)

Number re- caught (n2)

Number marked in the re-catch (n3)

10

20

2

10

20

2

10

20

3

10

20

3

10

20

2

SAMPLE SIZE:

30

Number caught initially (n1)

Number re- caught (n2)

Number marked in the re-catch (n3)

10

30

3

10

30

4

10

30

4

10

30

3

10

30

4

Table 4: Table showing data collected for Sample Size 40.

SAMPLE SIZE:

40

Number caught initially (n1)

Number re- caught (n2)

Number marked in the re-catch (n3)

10

40

5

10

40

6

10

40

5

10

40

6

10

40

7

Table 5: Table showing data collected for Sample Size 50.

SAMPLE SIZE:

50

Number caught initially (n1)

Number re- caught (n2)

Number marked in the re-catch (n3)

10

50

9

10

50

9

10

50

7

10

50

7

10

50

8

PROCESSED DATA:

Population is calculated using the following equation:

Total Population = Number caught x Number re-caught

Number marked in the re-catch

Or: N = (n1 X n2) / n3 Where N =Population Estimate

n1 = Number caught initially

n2 = Number re-caught

n3 = Number marked in the re-catch.

For example,

In Sample Size 10, where n1= 10, n2= 10 and n3= 1

N = 10X10/1

=100

This has to be repeated for all five readings of n3 in sample size 10 and thereafter for rest of the sample sizes.

Sample Size

Population Estimate

(N)

Average N for sample sizes

Standard Deviation

(±)

10

100.00

100.00

100.00

50.00

100.00

90.00

22.36

20

100.00

100.00

66.60

66.60

100.00

86.60

18.29

30

100.00

75.00

75.00

100.00

75.00

85.00

13.69

40

80.00

66.60

80.00

66.60

57.10

70.00

9.87

50

55.50

55.50

71.40

71.40

62.50

63.30

7.96

Table 6: Table showing Average N and SD for all sample size

Graph 1:

Graph 2:

Discussion:

The simplest of the mark-recapture methods is the Lincoln-Peterson index (Lincoln, 1930). A sample of the population is taken, the animals are marked, released, and a second sample is taken. After their initial release a proportion of the population will be marked. The proportion of marked animals in the second sample should reflect the proportion of animals marked in the entire population. This method helps in estimating and calculating the population size of any natural or other habitat without needing to count the species individually. Through this experiment, the Lincoln Index is simulated to study how the sample size may affect the accuracy of the population estimate. In the raw data, the n3, which is the number marked in the re-catch is shown to be increasing in each of the sample sizes. While in sample size, where 10 samples were marked and released, and 10 recaptured, the readings show that the recaptured numbers are 1,1,1,1 and 2. While in sample size 50 around 9 were the highest number of marked samples in the recapture in the 1st attempt. This indicates that evidently, as the sample size increases, the number marked in the recapture is likely to be higher. With the help of the raw data, after calculating the population estimate (N), the averages show a trend. It can be observed that as the sample size is increasing, the population estimate is decreasing. While the average population estimate for sample sizes 10, 20, 30, 40 and 50 are 90, 86.6, 85, 70 and 63.3 respectively, a decreasing trend can be observed in the calculated data. This suggests that the accuracy is improving with the increase in sample size, as the value of the population estimate is decreasing and therefore coming closer to the actual population value. This is supported by the calculation of the Standard Deviation of the averages of each of the sample sizes. Study proposes that as SD decreases, the accuracy of the required value becomes higher. This is because lower SD suggests less deviation, which results in a more accurate value of the population. The sample sizes 10, 20, 30, 40 and 50 recorded SD values of 22.36, 18.29, 13.69, 9.87 and 7.96 respectively. This declining trend of the SD values indicates that as the sample size is increasing, the SD value decreases due to less deviation, and therefore being more accurate in calculating the actual population size.

Evaluation:

Since the experiment is only a simulation of Lincoln Index, there would be a limitation in acquiring an exact population estimate. This is because the actual species in a natural environment, when captured, marked and released have other factors affecting the count. Factors such as natural calamities, death of a particular animal, type of marking that has a possibility of being erased. These factors do not affect the species (taken as papers) in the experiment. Therefore, more realistic simulations of animals can be taken.

Counting the re-captured and marked species, is a very time-consuming activity especially when done manually. As the sample size increases, it becomes more difficult to count all the pieces of paper. It poses as a limitation if this experiment needs to be carried out on a large scale. Thus, smaller sample sizes should be considered and other methods of counting (mechanically) can be practiced.

Conclusion:

The processed data and the results collected, through this experiment match the hypothesis. With the simulation of a natural habitat and species, the experiment helps in concluding that the sample size does affect the accuracy. If a higher sample size is taken, it is more likely to achieve a more accurate result (i.e. population size). Even in a natural habitat, this notion can hold the same result. The data collected also reflects the hypothesis, showing the Population Estimate averages decreasing, therefore being closer to the results and being more accurate. And the decrease in SD (standard deviation) also supports the claim, as it shows that as the sample size get bigger, the SD decreasing, therefore proving that it is closer to accuracy.