clearly to analyse and express the reasons why they advocated the mean as the safest course. But in the last two centuries this apparently simple question of the mean has been found to afford a field for the exercise of the utmost mathematical skill. Roger Cotes, the editor of the Principia, appears to have had some insight into the value of the mean; but profound mathematicians such as De Moivre, Daniel Bernoulli, Laplace, Lagrange, Gauss, Quetelet, De Morgan, Airy, Leslie Ellis, Boole, Glaisher, and others, have hardly exhausted the subject.
Several uses of the Mean Result.
The elimination of errors of unknown sources, is almost always accomplished by the simple arithmetical process of taking the mean, or, as it is often called, the average of several discrepant numbers. To take an average is to add the several quantities together, and divide by the number of quantities thus added, which gives a quotient lying among, or in the middle of, the several quantities. Before however inquiring fully into the grounds of this procedure, it is essential to observe that this one arithmetical process is really applied in at least three different cases, for different purposes, and upon different principles, and we must take great care not to confuse one application of the process with another. A mean result, then, may have any one of the following significations.
(1) It may give a merely representative number, expressing the general magnitude of a series of quantities, and serving as a convenient mode of comparing them with other series of quantities. Such a number is properly called The fictitious mean or The average result.
(2) It may give a result approximately free from disturbing quantities, which are known to affect some results in one direction, and other results equally in the opposite direction. We may say that in this case we get a Precise mean result.
(3) It may give a result more or less free from unknown and uncertain errors; this we may call the Probable mean result.
Of these three uses of the mean the first is entirely different in nature from the two last, since it does not yield an approximation to any natural quantity, but furnishes us with an arithmetic result comparing the aggregate of certain quantities with their number. The third use of the mean rests entirely upon the theory of probability, and will be more fully considered in a later part of this chapter. The second use is closely connected, or even identical with, the Method of Reversal already described, but it will be desirable to enter somewhat fully into all the three employments of the same arithmetical process.
The Mean and the Average.
Much confusion exists in the popular, or even the scientific employment of the terms mean and average, and they are commonly taken as synonymous. It is necessary to ascertain carefully what significations we ought to attach to them. The English word mean is equivalent to medium, being derived, perhaps through the French moyen, from the Latin medius, which again is undoubtedly kindred with the Greek μεσος. Etymologists believe, too, that this Greek word is connected with the preposition μετα, the German mitte, and the true English mid or middle; so that after all the mean is a technical term identical in its root with the more popular equivalent middle.