SQUARES, METHOD OF LEAST, an arithmetical process of great importance for combining observations, or sets of observations, so as to obtain the most probable value of a quantity which de pends on these observations. It is in fact the scientific method of taking certain averages, and it finds its most constant use in astronomy and other physical sciences. The necessity for applying the method arises from the fact that when the greatest precision of measurement is sought, repeated measurements of the same quantity do not agree. Thus, the altitude of a star at culmination, if care fully measured night after night by the same observer through the same instru ment, will in general come out a little different in the different observations. All the measurements will, however, lie within a certain range of variation; and if all are equally trustworthy, the arith metical mean will give the most proba ble value of the real altitude. The differences between this mean and the individual measurements on which it is founded are called the residuals. The important mathematical property of these residuals is that the sum of their squares is less than the sum of the squares of the differences between the individual measurements and any other single quan tity that might be taken. Now, this
principle of "least squares" holds not only for the simple case just described, but also for more complicated cases in which one observed quantity (y) is to be expressed as an algebraical function of another or of several independently observed quantities (x). Here the object is to f nd the most probable values of the assumed constants or parameters which enter into the formula. When these values are calculated we can calculate in terms of them and the observed x's a value of y corresponding to each set of observations. Comparing the calculated y's with the observed y's, we get a set of residuals the sum of whose squares is a minimum if the parameters have been calculated according to a particular process. It is this process which is de scribed as the method of least squares. Its basis is found in the mathematical principles of probability.