CHAPTER XVIL
THE SIMPLER CASES OF SAMPLING FOR VARIABLES:
PERCENTILES AND MEAN.
1-2. The problem of sampling for variables; the conditions assumed—
3. Standard error of a percentile—4. Special values for the percentiles
of a normal distribution—5. Effect of the form of the distribution
generally—6. Simplified formula for the case of a grouped frequency-
distribution—7. Correlation between errors in two percentiles of the
same distribution—8. Standard error of the interquartile range for the
normal curve—9. Effect of removing the restrictions of simple sampling,
and limitations of interpretation —10. Standard error of the arithmetic
mean—11. Relative stability of mean and median in sampling—12.
Standard error of the difference between two means—13. The tendency
to normality of a distribution of means—14. Effect of removing the re-
strictions of simple sampling—15. Statement of the standard errors of
standard-deviation, coefficient of variation, correlation coefficient and
regression, correlation-ratio and criterion for linearity of regression—186.
Restatement of the limitations of interpretation if the sample be small.
1. Iv Chapters XIIL.-XVI. we have been concerned solely with
the theory of sampling for the case of attributes and the frequency-
distributions appropriate to that case. We now proceed to
consider some of the simpler theorems for the case of variables
(¢f. Chap. XIII § 2). Suppose that we have a bag containing a
practically infinite number of tickets or cards bearing the recorded
values of some variable X, and that we draw a ticket from this
bag, note the value that it bears, draw another, and so on until
we have drawn n cards (a number small compared with the whole
number in the bag). Let us continue this process until we have
& such samples of n cards each, and then work out the mean,
standard-deviation, median, etc., for each of the samples. No one
of these measures will prove to be absolutely the same for every
sample, and our problem is to determine the standard-deviation
that each such measure will exhibit.
2. In solving this problem, we must be careful to define
precisely the conditions which are assumed to subsist, so as to
realise the limitations of any solution obtained. These conditions
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