reached, we have a definite value for each quantity under given constant conditions, namely, the arithmetical mean, and the average of the variations or the probable error will give an index of the accuracy with which that value has been determined.
Deductions from Results.–Suppose we have such a value as just mentioned, i. e., the arithmetical mean, what are the conclusions to be drawn? In the first place we can foretell the average value and the probable variations from that value when the conditions of future measurements are exactly the same as those of the set made, or do not differ to a greater degree than is negligible. In the second place, presuming that the same probability relations exist in another set of measurements, we can be sure of obtaining results within a given limit of variation with a definite degree of probability. In the third place, if we have two sets of measurements we can determine within what limits and with what sureness the probability underlying the one is the same as that underlying the other. The formulas for these deductions have been worked out by Poisson (Recherches sur la probabilité des jugements) and have been illustrated by Lexis (Einleitung in die Theorie der Bevölkerungsstatistik, Ch. V).
Applications.–Nothing has been said in regard to how accurate the measurements are to be before we can apply the principles just mentioned. Nothing should be said except that, whenever measurements of any kind are made, the computation of the results must follow the laws laid down by the science of measurement. Whether the accuracy is to 10% or to 1100 of 1% is a matter of indifference for the calculations. The claim put forth by some psychologists that the lack of accuracy in the measurements justifies the presentation and lumping of the results without observance of the rules and without a statement of the characteristic variations, enables them to prove anything they please with their figures. One psychologist not long ago made his measurements in groups of twenty-five and then selected twenty of each group from which to compute the result. Concerning the accuracy of the
