Fishery bulletin of the Fish and Wildlife Service - NOAA

BIOLOGY OP THE ATLANTIC MACKEREL 197
TABLE 9.—Example of the computation of limits for 5 classes within each of which an equal number of
catches would be expected if the distribution of stage A eggs during cruise I conformed to a normal
frequency surface; and the actual and theoretical number of catches for these class limits
0.0
0.2
0.4
0.6
0.8
1.0
Total
1
Equal fifths, cumulative
2
Tabular
number of
catches expected,cumulative
161
302
463
604
785
3
Tabular
class limits
for catch
magnitudes
10,000
3,190
1,010
321
102
Si
4
Actual class
limits for
catch magnitudes
6,806
1,863
686
186
S3
20
5
Actual number
of
catches
2
1
4
2
0
0
6
Theoretical
number of
catches
2 of the example. It indicates that 755/1000 of the frequency surface is to be taken
into account. Then 755 is multiplied by the items in column 1 of the example, giving
the series of items in column 2. Successive differences in this series would represent
equal fifths of the frequency surface out to 755, but it is, of course, not necessary to
compute these differences. The corresponding catch magnitudes are secured by
entering table 8 in the column of "Number of catches, cumulative," and reading, by
graphical interpolation, from the column of "Magnitude of catch." This gives the
series of column 3 in the example. These represent the class limits within each of
which one-fifth of the catches would fall if the maximum and minimum had been
10,000 and 34, respectively, and the distribution of catch magnitudes conformed
perfectly to the distribution expected from a normal frequency surface. Since they
were, instead, 5,806 and 20, respectively, the factor 5,806/10,000 is used to convert
them from the tabular to the actual basis, giving the values in column 4 in the example.
Between each pair of successive figures there should be found, theoretically,
an equal number of catches of stage A eggs from cruise I. In the first column of
table 19, cruise I, the adjusted totals of individuals of stage A are given, and a count
of those lying between each pair of specified class limits gives the numbers in column
5 of the example. Since the total number of catches was 9, neglecting those below
20, the theoretical number for each class is 9/5, or 1.8, as given in column 6 of the
example.
When the same computations are performed for the stage A eggs of cruises II,
III, and IV, and the actual number of catches are added together, by classes,
there results the series of values given under the appropriate heading in the first line
of table 10. There are now enough items in each class to apply the x 2 test; and the
probability P, that random variation would exceed the actual variation, is found to
be 0.85. This value would appear to be rather high; but when the work is done for
the remaining stages up to 22 mm. with due regard to the necessity of having fewer
classes for the later stages in order to keep the numbers per class high enough to use
the x 2 test, it is found that the values of P are distributed almost exactly as would
be expected, for there are 7 of them below and 8 above 0.5, and the mean is 0.53.
Hence it must be concluded that the catch magnitudes of stages up to 22 mm. larvae
are related to each other quite as would be expected had these stages been distributed
in the sea in conformity with the normal frequency surface.
1.8
1.8
1.8
1.8
1.8
9.0