XVIL.—SIMPLER CASES OF SAMPLING FOR VARIABLES. 337 
in a very large sample,” the phrases “the standard-deviation of 
X in the original record,” “the form of the frequency-distribution 
in the original record”: but in very many, perhaps the majority 
of, practical cases the very question at issue is the nature of the 
relation between the distribution of the sample and the distribu- 
tion of the record from which it is drawn. As has already been 
emphasised in the passages to which reference is made above, no 
examination of samples drawn under the same conditions can 
give any evidence on this head. 
3. Standard Error of a Percentile.—Let us consider first the 
fluctuations of sampling for a given percentile, as the problem is 
intimately related to that of Chaps. XIII.-XIV, 
Let X, be a value of X such that pN of the values of X in 
an indefinitely large sample drawn under the same conditions lie 
above it and ¢V below it. 
If we note the proportions of observations above X, in samples 
of » drawn from the record, we know that these observed values 
will tend to centre round p as mean, with a standard-deviation 
Vpg/n. If now at each drawing, as well as observing the pro- 
portion of X's above X,, say p +9, for the sample, we also proceed 
to note the adjustment e required in X, to make the proportion 
of observations above X,+e¢ in the sample p, the standard- 
deviation of e€ will bear to the standard-deviation of 8 the same 
ratio that e on an average bears to 4. But this ratio is quite 
simply determinable if the number of observations in the sample 
is sufficiently large to justify us in assuming that § is small—so 
small that we may regard the element of the frequency curve 
(for a very large sample) over which X, + e ranges as approximately 
a rectangle. If this assumption be made, and we denote the 
standard-deviation of X in a very large sample by o, and the 
ordinate of the frequency curve at X, when drawn with unit area 
and unit standard-deviation by z,, 
e=".3 
2, 
Therefore for the standard-deviation of e or of the percentile 
corresponding to a proportion p we have 
A 
Iz, = > n a (1) 
4. If the frequency-distribution for the very large sample be a 
normal curve, the values of y, for the principal percentiles may be 
taken from the published tables. A table calculated by Mr 
Sheppard (Table IIL, p. 9, in Zables for Statisticians and Biomet- 
Ory 
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