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Volume 84, 1956-57
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Error and Bias of Sampling

It will be seen below that, in the assessment of seasonal production, the variability of the distribution of eggs is so great that only gross changes in sample counts will be of any significance. Nevertheless, since some of the techniques described may be adaptable at a later date to estimation at a higher level of precision, it is of some value to consider as far as possible the errors and biasses which may arise either from the initial sampling or from sub-sampling.

No attempt has been made to interpret the sampling data in absolute terms* but it is believed that the surface samples give at least an approximate index of the total number of eggs in the area represented. It is known from laboratory studies that the live snapper egg tends to float at or near the surface, and sub-surface hauls have confirmed the fact that eggs are rare or absent below a depth of 5 fathoms. It seems probable, in fact, that by far the greatest number are concentrated in the surface layer and that, in calm weather, eggs (except those recently released and still finding their way to the surface) would be so close to the surface that nearly all would be susceptible to capture by the high-speed sampler. Variable bias might arise with increasing roughness of the sea, since eggs would then tend to be distributed to a greater depth. If this were so, a horizontal sample would give a lower count owing to the greater vertical dispersal of the eggs. However, since the sampler was towed directly behind the ship it is probable that any variable turbulence due to weather conditions would be negligible compared with the constant turbulence caused by the wake of the ship. To test this hypothesis the data for November, 1950, in which a variety of weather conditions were experienced, has been tabulated for size of catch against state of sea in Table 1.

[Footnote] * Assuming complete filtration, the volume of water sampled in a tow of one nautical mile would be approximately 5,800 litres.

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Table 1—Sample Count and State of Sea.
State of Sea  Smooth Slight Moderate Total
Eggs per mile. f f χ2 f f χ2 f f χ2 f
0–9 10 11 49 0.193 14 17 23 0 606 14 9 28 2 401 38
10–99 9 6 65 0 830 11 9 98 0.104 2 5 37 2.115 22
100–999 6 6 05 0.000 10 9.07 0.095 4 4 88 0.159 20
1000- 1 1 81 0 362 4 2 72 0.602 1 1.47 0.150 6
26 26.00 1.385 39 39.00 1.407 21 21.00 4.825 86

Total χ2 = 7.617, d. f. = 6, P = 0.2–0.3.

Against the frequency, f, of catches of given size is given the frequency, f, which would be expected if catch were independent of state of sea, while χ2 gives a measure of the probability of greater differences between f and f occurring by chance. Individual values of χ2 are all small and total χ2 indicates a probability of between 20 and 30 per cent that greater deviations would occur by chance, so that there is no reason to believe from this evidence that sample counts are influenced by state of sea.

As regards variations in catching power between different samplers, either of the same or different models, it was invariably found that if two nets were towed simultaneously the catches when preserved settled to the same volume as far as could be determined by a graduated cylinder reading to the nearest 0.1 ml. Although such paired samples were compared only volumetrically, it was clear that such differences as did exist between counts was unlikely to be greater than would be expected by chance in a randomly distributed population. For instance, a sample of 5 ml volume (which was about average size) would contain about 10,000 snapper eggs. If these were taken from a randomly distributed population the standard deviation of sample counts would be equal to the square root of the mean—i.e., 100. Since a Poisson distribution with such a large mean approximates closely to normal, one in 20 samples would deviate from the mean by more than 200 eggs, the volume of which is about 0.1 ml. Any marked over-dispersion would be expected to result in differences greater than this occurring fairly frequently. This conclusion is an interesting one since it is a well-known fact that plankton, particularly when the density is high, are often non-randomly distributed even to the extent that two nets on opposite sides of the same ship may take entirely different samples. It would appear that if this is the case with snapper eggs the mixing effect of the screw produces a random distribution within the wake of the ship.* Such an effect could make an appreciable reduction in the sampling error in an over-dispersed population of eggs since it increases both the width and the depth of the body of water for which the sample is truly representative.

The above discussion, although it does not exhaust all possible sources of error, reveals no reason to doubt that the samples give an unbiassed estimate of the relative density of eggs in the horizontal plane. The sub-sampling technique, being more amenable to direct control, can be examined more critically.

Throughout the examination of samples the usual statistical precautions were taken to ensure that sub-samples provided unbiassed and suitably accurate estimates

[Footnote] * This phenomenon may merit further attention in that it provides an independent approach to one of the fundamental problems of plankton sampling: When net samples are taken from an apparently homogeneous water mass, is the non-random distribution of sample counts due to a natural aggregation of the plankton, or to variations in the volume of water sampled? Barnes and Marshall (1951) have used a pump technique to demonstrate fairly conclusively that aggregation does in fact occur, though there still exists the possibility that the pump is also subject to sampling variation. The comparison of net or pump samples in a water mass which has been artificially “randomized” might possibly dispel any remaining doubt on this question.

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of the samples. The following two experiments will indicate that these requirements have, in most cases, been fulfilled. In the first experiment the eggs, regardless of species, in an entire sample were counted by eye in the sub-sampling tube. Since the reduced volume toward the end made adequate stirring impossible, only the first ten sub-samples are recorded separately. The results are shown in Table 2.

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Table 2.—Sub-sample Counts, All Eggs.
Volume of Subsample
(millilitres)		 Subsample Count
5.9 142
6.0 172
6.1 166
5.9 164
5.8 130
6.0 156
6.0 150
5.7 146
6.1 140
6.0 161
59.5 1,527
Remainder 40.5 988
100.0 2,515

For deviation of counts from the observed mean (152.70) χ2 is 10.479 with 9 degrees of freedom, giving a probability of about 0.3. If allowance is made for the slightly different sample counts expected owing to variations in volume, χ2 is reduced to 8.909 with a probability of nearly 0.5. In neither case is there any significant departure from a Poisson distribution of counts and the variation in sub-sample volume could probably be ignored without introducing any appreciable error. Since the total count for 100 ml is 2515, the expected mean count for the mean sub-sample volume (5.95 ml) will be 149.64 (which does not differ greatly from the observed mean, 152 70). For departure of sub-sample counts from the expected mean χ2 is 11 319. With 10 degrees of freedom there is a probability of a little over 0.3 of a higher departure occurring by chance.

In the second experiment a smaller sample was sub-sampled and then completely enumerated in species (snapper and other) using both the sub-sampler and the counting dish. The results are shown in Table 3.

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Table 3—Subsample Counts, by Species.
Snapper Other Species
Observed Expected Observed Expected Total
180 187.77 36 28 23 216
155 155.60 24 23.40 179
173 170 38 23 25.62 196
187 180.81 21 27.19 208
Remainder 103 103.44 16 15.56 119
798 798 00 120 120.00 918

The proportion of snapper eggs for the sample is 86 93 per cent. From this figure the expected frequency of the two categories is computed and compared with the observed frequency. χ2 is 4.421 with 4 degrees of freedom, giving a probability of between 0.3 and 0.4.

From the two experiments and from other similar checks made during the investigation, it is believed that the sub-sample counts are randomly distributed and

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give an unbiassed estimate of the true sample mean. In the typical case where at least 200 snapper eggs were counted, the coefficient of variation of the sub-sampling estimate may be computed as follows:

Where n = total sub-sample count for snapper eggs.

s = standard error of n

C = coefficient of variation of n

= s/n

Since snapper egg counts are distributed as a Poisson series.

s2 = n

C = 1/√n

Since n ≥ 200.

C ≤ 1/√200.

= 7% (approximately)

This maximum value of the co-efficient of variation will be applicable to about 80% of the samples recorded. The remainder will have a co-efficient of variation of an order seldom exceeding 10%, with the exception of a few cases where snapper eggs formed only a small proportion (less than 25%) of the total of all species. In such cases the laborious procedure of sorting over 800 eggs would have been required in order to obtain 200 snapper. Since such samples were few and had little effect on the distribution pattern the standard of accuracy has been relaxed in these cases.

In both stages the sub-sample counts are relatively large figures, so that their distribution will approximate closely to normal, and fiducial limits or tests of significance based on the co-officient of variation are appropriate.

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