•^66 PROFESSOR K. PEARSON ON THE MATHEMATICAL THEORY 
whence squaring, summing for all possible values, and remembering the definitions ol 
standard deviation and correlation coefficient we find : 
0 
o- X'. 
+ ‘5)^ = .(xiv.), 
where stn nds for (t. Im^. 
But the left-hand side of this equation is known from the previous reductions, 
o'x',) having the values in Table I. ; has just been determined. Hence a 
knowledge of Vj.^ would enable us to find without the labour of further correlation 
tables. The values of £Cj, had of course been measured in order to find x\, 
x'o, and a“'g, so that all we recjuired were their frequency distributions. They were as 
follows :— 
Table IX.—-Table of Frequencies of x^. 
Magnitude in 
^ incdics. 
Observer 
Magnitude in 
inches. 
Observer 
1. 
2. 
O 
o. 
1. 
2. 
3. 
1-35 
1 
1 
3-00 
52 
39 
t 
31 
1-50 
5 
8 
6 
3-15 
31 
28 
30 
1-6.5 
8 
5 
6 
3-30 
20 
23 
20 
1-80 
G 
15 
12 
3 • 45 
21 
17 
16 
1-95 
16 
29 
21 
3-60 
12 
10 
10 
2-10 
29 
36 
39 
3 ■ 75 
10 
6 
10 
2-25 
42 
47 
49 
3-90 
3 
4 
3 
2-40 
41 
54 
60 
4-05 
2 
2 
o 
2-55 
56 
62 
63 
4-20 
3 
1 
2 
2-70 
75 
55 
56 
4-35 
-- 
1 
1 
2.-85 
68 
55 
62 
4-50 
— 
2 
1 
Here the unit of grouping is '15 half-inch, and a magnitude in covers all the 
frequency between m — '075 and?7i + '075 half-inches. From these data we deduced 
m^c and cr^ being in half-inch units. 
Table X. 
Quantity. 
1. 
2. 
3. 
Mean, lUx 
2-7216 
2-6379 
2-6445 
S.D., (Tx 
-4909 
-5253 
-5072 
mxJa-x = Vx 
-1804 
-1991 
-1918 
Since 
we liave 
and 
— C » ”k 4,^'w (XiiCTxl iix^ • 
. (XV.). 
