Another comparison of the law with observation was made by Quetelet, who investigated the errors of 487 determinations in time of the Right Ascension of the Pole-Star made at Greenwich during the four years 1836–39. These observations, although carefully corrected for all known causes of error, as well as for nutation, precession, &c., are yet of course found to differ, and being classified as regards intervals of one-half second of time, and then proportionately increased in number, so that their sum may be one thousand, give the following results as compared with what Quetelet’s theory would lead us to expect:—[284]
Magnitude of | Number of Errors | Magnitude of | Number of Errors | ||
by | by | by | by | ||
0·0 | 168 | 163 | – | – | – |
+0·5 | 148 | 147 | –0·5 | 150 | 152 |
+1·0 | 129 | 112 | –1·0 | 126 | 121 |
+1·5 | 78 | 72 | –1·5 | 74 | 82 |
+2·0 | 33 | 40 | –2·0 | 43 | 46 |
+2·5 | 10 | 19 | –2·5 | 25 | 22 |
+3·0 | 2 | 10 | –3·0 | 12 | 10 |
– | – | – | –3·5 | 2 | 4 |
In this instance also the correspondence is satisfactory, but the divergence between theory and fact is in the opposite direction to that discovered in the former comparison, the larger errors being less frequent than theory would indicate. It will be noticed that Quetelet’s theoretical results are not symmetrical.
The Probable Mean Result.
One immediate result of the Law of Error, as thus stated, is that the mean result is the most probable one; and when there is only a single variable this mean is found by the familiar arithmetical process. An unfortunate error has crept into several works which allude to this subject. Mill, in treating of the “Elimination of Chance,” remarks in a note[285] that “the mean is spoken of as if it were exactly the same thing as the average. But the mean, for purposes of inductive inquiry, is not the average, or arithmetical mean, though in a familiar illustration of the theory the difference may be disregarded.” He goes on to say that, according to mathematical principles, the most probable result is that for which the sums of the squares of the deviations is the least possible. It seems probable that Mill and other writers were misled by Whewell, who says[286] that “The method of least squares is in fact a method of means, but with some peculiar characters.... The method proceeds upon this supposition: that all errors are not equally probable, but that small errors are more probable than large ones.” He adds that this method “removes much that is arbitrary in the method of means.” It is strange to find a mathematician like Whewell making such remarks, when there is no doubt whatever that the Method of Means is only an application of the Method of Least Squares. They are, in fact, the same method, except that the latter method may be applied to cases where two or more quantities have to be determined at the same time. Lubbock and Drinkwater say,[287] “If only one quantity has to be determined, this method evidently resolves itself into taking the mean of all the values given by observation.” Encke says,[288] that the expression for the probability of an error “not only contains in itself the principle of the arithmetical mean, but depends so immediately upon it, that for all those magnitudes for which the arithmetical mean holds good in the simple cases in which it is principally applied, no other law of probability can be assumed than that which is expressed by this formula.”
The Probable Error of Results.
When we draw a conclusion from the numerical results of observations we ought not to consider it sufficient, in cases of importance, to content ourselves with finding the simple mean and treating it as true. We ought also to ascertain what is the degree of confidence we may place in this mean, and our confidence should be measured by the degree of concurrence of the observations from which it is derived. In some cases the mean may be approximately certain and accurate. In other cases it may really be worth little or nothing. The Law of Error enables us to give exact expression to the degree of confidence proper in any case; for it shows how to calculate the probability of a divergence of any amount from the mean, and we can thence ascertain the probability that the mean in question is within a certain distance from the true number. The probable error is taken by mathematicians to mean the limits within which it is as likely as not that the truth will fall. Thus if 5·45 be the mean of all the determinations of the density of the earth, and ·20 be approximately the probable error, the meaning is that the probability of the real density of the earth falling between 5·25 and 5·65 is 1/2. Any other limits might have been selected at will. We might calculate the limits within which it was one hundred or one thousand to one that the truth would fall; but there is a convention to take the even odds one to one, as the quantity of probability of which the limits are to be estimated.
Many books on probability give rules for making the calculations, but as, in the progress of science, persons ought to become more familiar with these processes, I propose to repeat the rules here and illustrate their use. The calculations, when made in accordance with the directions, involve none but arithmetic or logarithmic operations.
The following are the rules for treating a mean result, so as thoroughly to ascertain its trustworthiness.
1. Draw the mean of all the observed results.