12 
The Probable Error of a Mean 
Hence the standard deviation of the curve is l/\/n-3. The fourth moment 
coefficient is equal to 
n - 2 7i - 4 
n - 3 ■ - 5 
etc. X " cos"-- e t-AW* 0dO 
n — 
2 
n 
-4 
etc. X J 
n — 
3 
11 
-5 ■■ 
11 — 
2 
11 
-4 
2C«-2) 
n — 
3 
' n 
— 5 
11 - 3 
+ 1 = 
3 
{11 -'6) {11 -by 
The odd moments are of course zero as the curve is symmetrical, so 
— 5 n — o 
Hence as ii increases the curve approaches the normal curve whose standard 
deviation is l/Vu — 3. 
/Sa however is always greater than 3, indicating that large deviations ■ are more 
common than in the normal curve. 
N 8 6 4 2 
DiAGEAM II. Solid curve ?/ = — x — cosi" ^, .rj>i=ia,nd. 
Brolieii line curve ?/ = , — c , tiie normal cnive with *he same s.d. 
VO 
s 
N 
1 / 
N \ 
S 
N 
/ 
/ 
/ 
\\ 
\\ 
\\ 
S 
N . 
1 
\ 
\\ 
\\ 
3 
N 
1 / 
—ft 
// 
// 
\\ 
\\ 
\\ 
S 
N 
// 
/ / 
/ / 
/ / 
\ 
\\ 
\\ 
\\ 
\\ 
3 
7^ 
/ / 
— 
\ \ 
1-5S 
OS -55 lOS V5S 
]Jist(i)ice of mean from mean of population 
I have tabled the area for the normal curve with standard deviation l/V? so as 
to compare with my curve for » = 10*. It will be seen that odds laid according 
to either table would not seriously ditfer till we reach z = "8, where the odds are 
about 50 to 1 that the mean is within that limit : beyond that the normal curve 
gives a false feeling of .security, for example, according to the normal curve it is 
99,986 to 14 (say 7000 to 1) that the mean of the population lies between — oo 
and + l-3s whereas the real odds are only 99,819 to 181 (about 5.50 to 1). 
* See p. 19. 
