342 THEORY OF STATISTICS.
percentiles in the same distribution, the student must be care-
ful to note that the errors in two such percentiles are not
independent. Consider the two percentiles, for which the values
of p and ¢ are p, q,, p, 9, respectively, the first-named being the
lower of the two percentiles. These two percentiles divide the
whole area of the frequency curve inte three parts, the areas of
which are proportional to ¢;, 1 — ¢; —p,, and p,. Further, since
the errors in the first percentile are directly proportional to the
crrors in ¢,, and the errors in the second percentile are directly
proportional but of opposite sign to the errors in p,, the corre-
lation between errors in the two percentiles will be the same as
the correlation between errors in ¢; and p, but of opposite sign.
But if there be a deficiency of observations below the lower
percentile, producing an error §, in ¢;, the missing observations
will tend to be spread over the two other sections of the curve
in proportion to their respective areas, and will therefore tend to
produce an error
3,= 2%, 5,
in p,. If then » be the correlation between errors in ¢, and p,,
¢ and e, their respective standard errors, we have
r2= _P2
& py
Or, inserting the values of the standard errors,
8 al
991
The correlation between the percentiles is the same in magni-
tude but opposite in sign : it is obviously positive, and consequently
correlation between errors | _ z NL Poth
; } - 3)
in two percentiles Gol
If the two percentiles approach very close together, ¢, and g,,
p; and p, become sensibly equal to one another, and the correla-
tion becomes unity, as we should expect.
8. Let us apply the above value of the correlation between
percentiles to find the standard error of the semi-interquartile
range for the normal curve. Inserting ¢;=p,=%, ¢,=p,=4%, we
find r=1. Hence the standard error of the interquartile range
is, applying the ordinary formula for the standard-deviation of a
difference, 2/,/3 times the standard error of either quartile, or
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