we get n different results, provided we make the unit of measurement fine enough. If by x we denote the variations from the arithmetical average, and if n be infinitely large, then the variations will occur with probabilities according to the well-known law,
y = ke−h2x2dx,
provided we make one of two suppositions: (1) the single variations are made up of small elementary independent variations, which are equally likely to be positive or negative; (2) the most probable value is the arithmetical mean. The former is the supposition of Laplace and Hagen; the latter is that of Gauss.
Neither of these suppositions is allowable in psychological measurements, or in physical ones, either, except as furnishing results sufficiently accurate. That they have justified themselves in physics is due to the facts: (1) that in all physical measurements the surrounding conditions are kept in a high degree of constancy; (2) that in all judgments in regard to the accuracy of physical work we presuppose that there were no sources of error comparable in magnitude with the measure of precision. Under such circumstances the occurrence of the elementary errors (or variations) in groups would have Comparatively little effect, and we can suppose them to be independent. In psychology the case is different. We cannot yet get our conditions so completely under control as in physics; the state of affairs somewhat resembles that in statistics. We are not justified in supposing that the variations are independent;[1] on the contrary, from the very large and irregular mean variations that we obtain, from our experience in gradually eliminating sources of error, and from our knowledge of varying circumstances that we cannot eliminate or measure, we know that the variations must occur in groups. The variations will therefore not follow the law of probability, and the arithmetical mean may or may not be the most probable value. A critical treatment of the variations, their signs, their successive differences, and the signs of the
- ↑ Cattell, On Errors of Observation, Am. Jour. of Psychol., 1893, V. 287.
