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for illustration, that the region is as given in Table 13.1, and that this region of

A=144 km^2 is to be sampled by n=4 transects each of area a=12 km^2. Sampling

intensity is hence na/A= 4 ×12 /144 =0.333.

In Table 13.1 the rows represent transects and the marginal totals the number of

animals on each transect. Numbering the transects from 1 to 12 and selecting at

random with replacement from this set we draw transect numbers 4, 8, 1, and 4. On

surveying these transects we would obtain counts of

Transect: 1 4 4 8

Count: 142 149 149 127

Note that transect number 4 has been drawn twice, so in practice we survey only

three transects although the count from transect number 4 enters the calculation twice.

Density is estimated as the sum of the transect counts (142 + 149 + 149 +127)

divided by the sum of the transect areas (12 + 12 + 12 +12). Thus:

D=∑y/∑a=567/48 =11.81/ km^2

The precision of that estimate is indexed by its standard error SE(D), which is

itself an estimate of what the standard deviation of many independent estimates of

density would be, each estimate derived from four transects drawn at random with

replacement:

SE(D) =1/a×√[(∑y^2 −(∑y)^2 /n)/(n(n−1))]

That is a slight approximation. To be exactly unbiased it should be multiplied by a

further term √[1 −(∑a)/A], but that usually makes so little difference that it tends

to be ignored.

The calculation tells us that this hypothetical distribution of estimates, each of

them made in the same way as we made ours, with the same sampling frame and

the same sampling intensity, only the draw of sampling units being different, would

have a standard deviation in the vicinity of ±0.43. In fact, that is likely to be

an underestimate because it is based on only four sampling units, three degrees of

freedom. With samples above 30 sampling units we can form 95% confidence

limits of the estimate by multiplying by 1.96, but for smaller samples we must choose

a multiplier from a Student’s t-table corresponding to a two-tailed probability of

0.05 and the degrees of freedom (d.f.) of our sample. In the case of d.f. =3, the

multiplier is 3.182 and so the 95% confidence limits of our estimate of density are

±3.18 ×0.43 =±1.37.

The number of animals (Y) in the surveyed region can now be calculated as the

number of square kilometers in that region (A) multiplied by the estimated mean

number per square kilometer (D):

Y=AD= 144 ×11.81 = 1701

It has a standard error of:

SE(Y) =±A×SE(D) =± 144 ×0.43 =± 62

228 Chapter 13