XVIL.—SIMPLER CASES OF SAMPLING FOR VARIABLES. 339
“peaked” distribution by superposing a normal curve with a
small standard-deviation on a normal curve with the same mean
and a relatively large standard-deviation. To give some idea of
the reduction in the standard error of the median that may be
effected by a moderate change in the form of the distribution, let
us find for what ratio of the standard-deviations of two such curves,
having the same area, the standard error of the median reduces to
o//n, where o is of course the standard-deviation of the compound
 distribution.
Let oy, 0, be the standard-deviations of the two distributions,
and let there be n/2 observations in each. Then
of +o}
g= v “5 (@)
On the other hand, the value of Y, 18—
IRE Mgr 1 Wo
22x. 0 22.0, 2
Hence the standard error of the median is
/ 2r S199 [AY
Von oy, + oy
(¢) is equal to o/In if
(01+ 03) Voitai_,
2 roe, :
Writing oy/o =p, that is if
(Lp) JT+p2_,
2 Amp
P +203 + (2 - 4m)p2 + 2p +1 =0.
This equation may be reduced to a quadratic and solved by
1
taking p + 28 & new variable. The roots found give p=2-2360
+v..0r 04472... the one root being merely the reciprocal of
the other. The standard error of the median will therefore be
/y/n, in such a compound distribution, if the standard-deviation
of the one normal curve is, in round numbers, about 2} times
that of the other. If the ratio be greater, the standard error
of the median will be less than a/n/n. The distribution

or
(e,
\S,