H. E. Soper 
111 
needing to be accounted for. The individual irregularities in the observations are 
more than would be expected in random samples of homogeneous material and it 
is probable that these jumps which make a good fit of any continuous curve what- 
ever an impossibility are partly due to grouping. If the grouping of the original 
material were too coarse there would be a tendency in small samples for statistical 
constants to centre round certain values. Another possible source of error is in 
the mixing and in the drawing out of samples. Although a great deal of trouble 
was undoubtedly taken in these experiments, yet there always seems room for a 
little involuntary order in repetitions intended to go solely by chance. 
The curves were planimetered and the following tables show the comparison of 
theoretical with observed frequencies of the grade values. It will be seen that 
the differences are not systematic but that the + and — errors are fairly mixed. 
On calculating the square contingency, yf, and deducing from this and the number 
of groups, ri, the probability, P, of these differences being purely that of sampling* 
such probability comes out in most cases very small. It seems legitimate in this 
instance to see to what extent grouping will smooth down the irregularities and 
yet show the general resemblance and, with this in view, the differences in the 
columns headed e are calculated and the grouping therein shown may be taken to 
indicate what is necessary to bring the probabilities within reasonable distance of 
expectation. 
p = 0, n = 4. p = 0, n = 8. 
r 
Calculated 
frequency 
m 
Observed 
frequency 
Difference 
e 
c 2 
7)1 
Calculated 
frequency 
m 
Observed 
frequency 
Difference 
e 
e 2 
m 
•925—1-0 
27-94 
22-5 
5-44 
1-06 
•6 
•6 
•60 
•825— 
37-25 
31-5 
5-75 
•89 
41 
3 
1 
1 
■30 
•725— 
37-25 
24-0 
13-25 
4-71 
11-6 
12 
+ 
•4 
01 
•625— 
37-25 
35-0 
2 - 25 
•14 
20-6 
11 -5 
9 
•1 
4 
•02 
•525— 
)) 
34-0 
3-25 
•28 
31-9 
28-5 
3 
4 
■36 
•425— 
47-0 
+ 
9-75 
2-55 
42-3 
46 
+ 
3 
7 
■32 
•325— 
» 
30-5 
6-75 
1-22 
51-6 
47-5 
4 
1 
•33 
•225- 
» 
46-5 
+ 
9-25 
2-30 
59-9 
70 
+ 10 
1 
1 
■70 
•125— 
?» 
44-0 
+ 
6-75 
1-22 
65-8 
57-5 
8 
3 
1 
05 
•025— 
>J 
32-0 
5-25 
•74 
69-4 
70 
+ 
6 
01 
1-925— 
>> 
45-0 
+ 
7-75 
1-61 
70-3 
60-5 
9 
8 
1 
•37 
1-825— 
•>•> 
43-0 
T 
5-75 
•89 
67-9 
71-5 
+ 
3 
6 
19 
1-725— 
1J 
41-0 
+ 
3 75 
•38 
63-1 
76 
+ 12 
9 
2 
•64 
1-625— 
)J 
37-0 
■25 
•00 
56-3 
63 
+ 
6 
7 
80 
1-525— 
5) 
44-0 
+ 
6-75 
1-22 
47-1 
42 
5 
1 
•55 
1-425— 
)5 
40-0 
+ 
2-75 
•20 
36-9 
33 
3 
9 
•41 
1-325— 
5> 
38-5 
+ 
1-25 
•04 
26-1 
29 
+ 
2 
9 
32 
1-225— 
!> 
32-5 
4-75 
•61 
15-3 
20 
+ 
4 
7 
1 
44 
1-125— 
36-0 
1-25 
•04 
7-4 
8 
+ 
6 
•05 
1—1-125 
46-56 
41-0 
5-56 
•83 
1-8 
1 
8 
36 
745 20-93 750 16'83 
n' = 20, % == 20-93, P = -.341. n' = 20, % 2 = 1683, P = -601. 
* Tables for testing the goodness of fit, W. Palin Elderton, Biometrika, Vol. i. p. 155. 
