Elkanor Pairman and Karl Pearson 
243 
good results it is quite unnecessary for the terminal ordinates and differential 
coefficients of the actual curve of frequency and the auxiliary terminal curve to be 
closely identical. 
(7) Illustration II. Now let us take a normal curve containing 1,000,000 
individuals with a standard deviation of unity, and let us suppose the frequency 
grouped on 0'5 x standard deviation subranges, the first such subrange being central. 
Then, adjusting to units, we have the following system : 
- -25— 
+ -25 
197,414 
+ -25— 
+ -75 
174,666 
+ ^r,— 
+ 1-2.5 
120,977 
+ 1-25-- 
+ 1-75 
65,591 
+ 1-75— 
+ 2-2.5 
27,834 
+ 2-25— 
+ 2-75 
9,245 
+ 2-75— 
+ 3-25 
2,402 
+ 3-2.'5— 
+ 3-75 
489 
+ 3-75_ 
+ 4-25 
78 
+ 4-25— 
+ 4-75 
10 
+ 4-75— 
+ 5-25 
1 
To test the error introduced by our adjustments, take second moments for the 
complete curve about the centre of the group from — -25 to + '25. We have 
vl = 0, v.: = 4-083,394. 
Using Sheppard's correction as abruptness coefficients are zero, we have 
fjio = 4-000,061 in working units, 
= 1-000,0152 in actual units. 
Accordingly o- = 1-000,008, which is a quite good approximation to unity. The 
error introduced by our adjustments for omitted decimals is therefore not great. 
(a) We will start first with the singly truncated normal curve given below 
and /^, = //, i.e. the area from ,r= 1*25 onwards. 
Moments about 
stump 
r'i'= 1-029,.51.3 
1'./= 1-693,994 
„.,'= 3-883,416 
!,,,'= 10-974,937 
Total 105,650, 
(ill working units) 
and determine its moment-coefficients, as a frequency curve having high contact at 
one end and marked abruptness at the other. In this case all the b's are zero and 
we only need to find the as. Clearly «' = a. 
