6 
The Prohahle Error of a Mean 
By continuing this process we tind 
o , m - 2 
or 
/„_, = (-) (n-S)(«-5)...3.1/„ 
^ 2 (»_3)(«-5)...4.2/i 
according as ?(, is even or odd. 
But /o is 
and /i is 
Hence if n be even, 
..•=x ^2 
xe'^'dx = 
e -"■ 
J 0 
n 
x=0 M 
Area 
and if n be odd 
(.-:3)(H-5)...3.1^|(^ 
Area 
2 
C« -3)(«-5)...4. 2 
Hence the equation may be written 
N 
(/i.-3)(n-5)...:3. 1 V ttW 
2 /?i 
- 2 e ('(I even) 
or 
N 
(?i-3)(?i-5)...4.2 W 
where N as usual represents the total frequency. 
2 ^71-2 g 2<r'' o(Jd) 
Section 11. 
To show that there is no correlation between (a) the distance of the mean of 
a sample from the mean of the population and (b) the standard deviation of a 
sample with normal distribution. 
(1) Clearly positive and negative positions of the mean of the sample are 
equally likely, and hence there cannot be correlation between the absolute value 
of the distance of the mean from the mean of the population and the standard 
deviation, but (2) there might be correlation between the square of the distance 
and the square of the standard deviation. 
Let i(- = 
C'^]' and ,. = ^-i) -(«(?■>)' 
\ 11 J 11 \ n / 
Then if m/, if/ be the mean values of u" and we have by the preceding 
, , , (n — 1 ) , , u., 
part ilf J = uo and = . 
