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 com- 
pound 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,