18 The Probable Error of a Mean 
Mean value of standard deviations ; calculated 2'186 ± '023 
„ ,, „ observed 2'179 
Difference = - "007 
Standard deviation of standard deviations : — 
Calculated "9224 ± -016 
Observed '9802 
Difference = + -0578 
Comparison of Fit. Theoretical Equation: y = — t= x^e . 
Scale in terms 
of standard 
deviation of 
population 
o 
0) 
o 
• 
o 
o 
o 
o 
00 
o 
4J 
91 
0 
4J 
00 
•9 to 1 -0 
0 
4J 
so 
•-I 
0 
©( 
»M 
0 
+j 
so 
>~i 
C 
4J 
0 
1-6 to 1-7 
greater 
than 1 -7 
Calculated 
frequency 
u 
27 
64i 
78| 
87 
88 
811 
71 
58 
45 
33 
23 
15 
H 
7 
Observed 
frequency 
2 
14 
51 
64| 
91 
94| 
684 
73 
48| 
40| 
421 
20 
22| 
12 
5 
7J 
Difference 
+ i 
+ 
+4 
+ 
+ 12i 
+ 7^ 
-16 
+ 2 
-9h 
-4i 
+ 9^ 
-3 
+ 7* 
+ 21 
1 
2 
+1 
whence ;^2 = 21-80, P=-19. 
Calculated value of standard deviation 1 (+ •017) 
Observed „ „ „ -982 
Difference = -^8 
2 
Comparison of Fit. Theoretical Equation: y = - cos^d, 2 = tan ^. 
Scale of z 
so 
1 
i3 
+3 
Oi 
.2, 
\ 
0 
so 
1 
to 
1 
4J 
1 
>J0 
1 
-is 
>0 
LO 
1 
1 
0 
UO 
1 
uo 
-<!- 
1 
c 
l-O 
1 
1 
0 
1 
+ 
0 
1 
.0 
+ 
c 
»i0 
+ 
^"^ 
+ 
0 
+ 
uo 
+ 
c 
4 
1*0 
uo 
+ 
0 
'-I 
+ 
UO 
+ 
0 
-fj 
LO 
+ 
s'o 
+ 
0 
-+J 
+ 
more than +3-05 
Ccilculated 
frequency 
5 
!)i 
13i 
34^ 
44i 
78^ 
119 
141 
119 
784 
44i 
131 
9| 
5 
Observed 
frequency 
4 
15i 
18 
33i 
44 
75 
122 
138 
120^ 
71 
461 
36 
11 
9 
6 
Difference 
-1 
+ 6 
+ 4i 
-1 
1 
-3i 
+ 3 
-3 
+ I5 
-7* 
+ 2 
+u 
-2i 
1 
~ "5 
+ 1 
whence ;^2 = 7-39, /'=-92. 
A very close fit. 
We see then that if the distribution is approximately normal our theory gives 
us a satisfactory measure of the certainty to be derived from a small sample in 
both the cases we have tested ; but we have an indication that a fine grouping is 
