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 :

1

by Theory.

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 indiIt will be noticed that Quetelet's theoretical results are not symmetrical.

cate.

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 2 that "the mean is spoken of

1 Quetelet, Letters on the Theory of Probabilities, translated by Downes, Letter XIX. p. 88. See also Galton's Hereditary Genius, p. 379. System of Logic, bk. iii. chap. 17, § 3. 5th ed. vol. ii. p. 56.

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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 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,2" 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, 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

1 Philosophy of the Inductive Sciences, 2nd ed. vol. ii. pp. 408, 409. 2 Essay on Probability, Useful Knowledge Society, 1833, P. 41. 3 Taylor's Scientific Memoirs, vol. ii. p. 333.

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 525 and 565 is. 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.

2. Find the excess or defect, that is, the error of each result from the mean.

3. Square each of these reputed errors.

4. Add together all these squares of the errors, which are of course all positive.

5. Divide by one less than the number of observations. This gives the square of the mean error.

6. Take the square root of the last result; it is the mean error of a single observation.

7. Divide now by the square root of the number of

observations, and we get the mean error of the mean result.

8. Lastly, multiply by the natural constant o6745 (or approximately by 0674, or even by ), and we arrive at the probable error of the mean result.

Suppose, for instance, that five measurements of the height of a hill, by the barometer or otherwise, have given the numbers of feet as 293, 301, 306, 307, 313; we want to know the probable error of the mean, namely 304. Now the differences between this mean and the above numbers, paying no regard to direction, are 11, 3, 2, 3, 9; their squares are 121, 9, 4, 9, 81, and the sum of the squares of the errors consequently 224. The number of observations being 5, we divide by 1 less, or 4, getting 56. This is the square of the mean error, and taking its square root we have 7:48 (say 7), the mean error of a single observation. Dividing by 2:236, the square root of 5, the number of observations, we find the mean error of the mean result to be 335, or say 33, and lastly, multiplying by 6745, we arrive at the probable error of the mean result, which is found to be 2259, or say 21. The meaning of this is that the probability is one half, or the odds are even that the true height of the mountain lies between 301 and 306 feet. We have thus an exact measure of the degree of credibility of our mean result, which mean indicates the most likely point for the truth to fall

upon.

The reader should observe that as the object in these calculations is only to gain a notion of the degree of confidence with which we view the mean, there is no real use in carrying the calculations to any great degree of precision; and whenever the neglect of decimal fractions, or even the slight alteration of a number, will much abbreviate the computations, it may be fearlessly done, except in cases of high importance and precision. Brodie has shown how the law of error may be usefully applied in chemical investigations, and some illustrations of its employment may be found in his paper.1

The experiments of Benzenberg to detect the revolution of the earth, by the deviation of a ball from the perpen

1 Philosophical Transactions, 1873, p. 83.

dicular line in falling down a deep pit, have been cited by Encke as an interesting illustration of the Law of Error. The mean deviation was 5 086 lines, and its probable error was calculated by Encke to be not more than 950 line, that is, the odds were even that the true result lay between 4136 and 6036. As the deviation, according to astronomical theory, should be 46 lines, which lies well within the limits, we may consider that the experiments are consistent with the Copernican system of the universe.

It will of course be understood that the probable error has regard only to those causes of errors which in the long run act as much in one direction as another; it takes no account of constant errors. The true result accordingly will often fall far beyond the limits of probable error, owing to some considerable constant error or errors, of the existence of which we are unaware.

Rejection of the Mean Result.

We ought always to bear in mind that the mean of any series of observations is the best, that is, the most probable approximation to the truth, only in the absence of knowledge to the contrary. The selection of the mean rests entirely upon the probability that unknown causes of error will in the long run fall as often in one direction as the opposite, so that in drawing the mean they will balance. each other. If we have any reason to suppose that there exists a tendency to error in one direction rather than the other, then to choose the mean would be to ignore that tendency. We may certainly approximate to the length of the circumference of a circle, by taking the mean of the perimeters of inscribed and circumscribed polygons of an equal and large number of sides. The length of the circular line undoubtedly lies between the lengths of the two perimeters, but it does not follow that the mean is the best approximation. It may in fact be shown that the circumference of the circle is very nearly equal to the perimeter of the inscribed polygon, together with one-third part of the difference between the inscribed and circumscribed polygons of the same number of sides. Having

1 Taylor's Scientific Memoirs, vol. ii. pp. 330, 347,

&c.