244 
PROFESSOR K. PEARSON ON THE MATHEMATICAL THEORY 
pricking with a needle point. This was done in triplicate by running the needle 
point through three adjusted sheets. These segments formed a random distribution 
of lengths placed on a series of horizontal lines. Each observer now took 500 such 
lines—the series being the same for each—struck a pencil stroke with a fine pencil 
through the needle points terminating each segment, and then bisected that segment 
with a third pencil stroke at sight. AVe thus obtained three series of estimates of 
the midpoints of the same gi’oipj of lines by three apparently independent observers. 
The judgments were made in the same room, under practically the same conditions of 
light for each individual, but each experimenter was not necessarily bisecting the same 
line at the same instant of time. The common factors were the length of the line 
and its position relative to the edge of the })aper, which latter varied from line to 
line. It does not appear to me that these factors are more or less influential than the 
sameness of influences which must ever arise when two or more individuals judge the 
same phenomenon. 
The actual length of the lines and the distance from the left-hand terminal of the 
point guessed as mldpoiiit were now very carefully measured ; whatever errors occur in 
these measurements, and of course such must exist, they are of a totally different order 
of magnitude to the errors of midpoint judgment.^ The letter u will be used to denote 
the length of any line, x for the distance from the left-hand terminal to the experi¬ 
mental bisection, x = x — will stand for the error in placing the midpoint, 
considered })ositive when towards the right. The subscript 1 refers to l)r. Lee’s 
judgment, the subscript 2 to my judgment, and the subscript 3 to Mr. Yule’s 
judgment. I should have liked to have taken 1000 instead of 500 judgments, but the 
labour of experimenting, and especially also of arithmetical reduction is so great that 
we had to limit ourselves to the smaller number. Even that, I believe, is far greater 
than has yet been used in the determination of personal eijuation. 
ri ijriori, it seemed reasonable to me that the longer the line the greater would be 
the error of its bisection. Accordingly x ju, or the I'atio of the error to the length of 
the line, was taken in the first place as the quantity to be tabulated. I call this 
quantity X'. Dr. Lee spent several months of the summer of 1896 in the reduction 
of the observations on this basis, and the series of diagrams giving the frequency 
curves were drawn for X'. The reduction, however, showed at once that the values 
of X' for different observers were correlated. Such cori'elation of what I then 
thought must -be independent judgments led me to more closely investigate the 
matter. I attributed this correlation of independent judgments to spurious correla¬ 
tion due to the use of indices, and I determined to reconsider the subject on an 
entirely different experimental plan, after develo 2 )ing the theory of spurious 
correlation, t 
* That judgment was made rapidly as soon as the Jieedle points terminating the line had been marked 
so as to be visible. 
t See ‘ Roy. Soc. Rroc.,’ vol. 60, p. 489. 
