252 
PKOFESSOR K. PEARSON OX THE MATHEMATICAL THEORY 
So far, then, as this first series of experiments goes, we have ample justification for 
grouping our deviations. 
(h.) Bright Line Series. 
In this case I compared the results for ungrouped and grouped observations not 
only as far as concerns absolute personal equations, but also for the relative personal 
equations, and even for the coefficients of coi'relation. We have therefore a still 
wider basis for drawing inferences. This is done in Table II., x being now the error, 
positive if the bright line is recorded on the strip as being below its true joosition. 
To find a length on the observation strip from that on the recording strip we have 
to multiply by the factor L734. 
Table II. 
519 obserA'ations. 
Absolute jAersonal equation. 
Relative personal equation. 
1 
Xo. 
•^'3- 
Xo — a's. 
~ -r'l. 
.-;i - X2. ' 
5Iean, ungrouped 
„ grouped . | 
+ -06724 
+ -03538 
+ -07774 
± -03521 
- 1-14906 
+ -03480 
- 1-14483 
± -03473 
- -48563 
+ -05377 
- -44635 
± -05393 
i - -66343 
+ -05170 
- -68275 
± -05148 
- -55287 
+ -05954 
- -61145 
± -05943 
+ 1-21630 
± -04928 
+ 1-21518 
± -04932 
S.D. ungrouped . 
„ grouped . | 
1-19495 
+ -02502 
1-18913 
± -02489 
1-17546 
+ -02461 
1-17289 
± -02455 
1-81599 
+ -03802 
1-82146 
± -03813 
1-74616 
+ -03656 
1-73883 
± -03640 
2-01091 
+ -04210 
2-00717 
± -04202 
1-66454 
+ -03485 
1-66597 
± -03488 
Correlations. 
I'is- 
rsi. 
ri2. 
Ph •23- 
P2) 31- 
P3) 12- 
Ungroujjed . . 
Grouped . . . -^ 
-3819 
+ -0253 
-3908 
+ -0251 
-1571 
+ -0289 
-1624 
± -0288 
-0139 
+ -0296 
-0051 
± -0296 
-5625 
+ -0202 
-5653 
+ -0201 
-3055 
+ -0268 
-3056 
+ -0268 
-6154 
+ -0184 
-6127 
+ -0185 
We see at once from this table that the probable errors of means, standard 
deviations, and correlations are for all practical purposes the same whether Ave group 
the observations or not. In the next place Ave find that, judged by these probable 
errors, the differences are less than Avould arise from the results of random sampling. 
Thus in all cases the difterences are less than the 2 )robable errors, and in most 
cases very considerably less. The greatest diAmrgence occurs in the relatiA*e personal 
equation of Dr. Lee and myself, I)ut eAmn in this case the difference is just less than 
the probable error. We may accordingly coiiclude that AA'ith such a number of 
