Page:Philosophical Review Volume 2.djvu/700

686 differences will show whether the measurements follow the law of probability or not. If they do not, we have no recourse except to empirical treatment.

Empirical Treatment of the Results.–When the usual treatment of our results is not applicable, we are forced to fall back on empirical methods. Let us take our n measurements, say of reaction-time, and lay off on the axis of abscissas values corresponding to the successive results obtained, e. g., 180σ, 181σ, 182σ,… and erect ordinates proportional to the number of times each value occurs. If the variations conformed to the suppositions mentioned above we would get a curve resembling the ordinary probability curve. What we actually do get, is a curve with several maxima instead of one; and the curve can be regarded as made up of several probability curves with different mean values and different degrees of steepness. This shows us that our measurements are running in groups, and that the factors going to influence the results are working in combinations. Our measurements were made under conditions that were not controlled so as to give a well defined result. In the measurements of simple reaction-time a curve with two maxima, say one much more prominent than the other, would show that what we had been measuring as simple reaction-time had not been well defined, that there was one form which had predominated and another form not so prominent. If we take the arithmetical mean of all the results we are averaging two different classes of things together. Exactly the same results are obtained in statistical measurements. The arithmetical mean has been found quite unsatisfactory; if we take the mean height of a community composed of part English and part French, we have a mixture of two groups and will get a curve of results with two maxima.

This indication of the grouping of variations leads to a further analysis of the quantity measured till the variations from the probability curve become small in comparison with the desired or the possible accuracy. When this point is