10 
The Prohable Error of a Mean 
Substituting this value of D we get 
3~ 
In 
1 - 
1 
V 
1 - 
1 
2/i 16?i2 
3<r* / 1 
A'4 = T-, 1 1 + 
1 
Consequently the value of the standard deviation of a standard deviation which 
we have found 
1 - 
becomes the same as that found for the normal 
curve by Professor Pearson {a/'\/2n) when n is large enough to neglect the l/4>n in 
comparison with 1. 
Neglecting terms of lower order than ~ we find 
13. 
2n - 3 
n (4h'-~3) ' 
Consequently as n increases /Sa very soon approaches the value 3 of the normal 
curve, but /3i vanishes more slowly, so that the curve remains slightly skew. 
Diagram I. Frequency curve giving the distribution of Standard Deviations of samples of 10 taken 
from a normal population. 
iV 10^ 
Equation y -- 
7.5,3 
10x2 
2cr2 
1-6 
1 
1-2 
% 

— - 
Popuit 
% 
N/ 
•S 
1= E \ 
'(7 
1 
it l| \ 
-i 

' ' k 
% 
Stam 
7^ 
Diagram I shows the theoretical distribution of the s.D. found from samples 
of 10. 
NW^ /2x^ -2^2 
Section V. 
?i — 2 n — 4 
Some properties of the curve y 
^4 2 . . , 
. - if n be even 
?i — 3 H — 5 ' " I r) 3 
,4' 2 
(1+^0"^ 
if n he odd 
Writinof z = tan 0 the equation becomes y = - — ^ . , . . . etc. x cos" 0, whicli 
affords an easy way of drawing the curve. Also dz = ddjcos' 6. 
