SUPPLEMENTS—GOODNESS OF FIT. R75
1 success, and the frequency of 12 successes with that of 11
successes.
In the second place, the proof outlined assumes that the
theoretical law is known a priorz. In a large number, perhaps
almost the majority, of practical cases in which the test is ap-
plied this condition is not fulfilled. We determine, for example,
the constants of a frequency curve from the observations them-
selves, not from a priori considerations: we determine the
“independence values” of the frequencies for a contingency
table from the given row and column totals, again not from
a priori considerations. This general case is dealt with below,
in the section headed * Comparison Frequencies based on the
Observations.”
Finally, attention should be paid to the run of the signs of
the differences m’—m. The method used pays no attention to
the order of these signs, and it may happen that x* has quite a
moderate value and P is not small when all the positive differences
are on one side of the mode and all the negative differences on the
other, so that the mean shows a deviation from the expected value
that is quite outside the limits of sampling, or that the differences
are negative in both tails so that the standard deviation shows
an almost impossible divergence from expectation. In the first
example on the preceding page all the differences are negative up to
5 successes, positive from 6 to 10 successes, and negative again for
11 and 12 successes. This is almost the first case supposed, and
in fact we have already found (p. 267) that the mean deviates
from the expected value by 5°1 (more precisely 5:13) times its stan-
dard error. From Table II. of Tables for Statisticians we have :—
Greater fraction of the area of a normal
curve for a deviation 5°13 . : . "9999998551
Area in the tail of the curve . . *01)00001449
Area in both tails . : . '0000002898
so that the probability of getting such a deviation (+ or —) on
random sampling is only about 3 in 10,000,000. The value found
for P (0015) by the grouping used is therefore in some degree
misleading. If we regroup the distribution according to the
signs of m"— m, we find
m—— sgt ved Expected
requency Frequency.
0-5 1426 1586
6-10 2659 2497
11-12 1 13
Total. 28 1095
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