The principles of science

by William Stanley Jevons

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 0·6745 (or approximately by 0·674, or even by 2/3), 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 1/2), 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 3·35, or say 3 1/3, and lastly, multiplying by ·6745, we arrive at the probable error of the mean result, which is found to be 2·259, or say 2 1/4. 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 3/4 and 306 1/4 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.‍[289]

The experiments of Benzenberg to detect the revolution of the earth, by the deviation of a ball from the perpendicular line in falling down a deep pit, have been cited by Encke‍[290] 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 4·136 and 6·036. As the deviation, according to astronomical theory, should be 4·6 lines, which lies well within the limits, we may consider that the experiments are consistent with the Copernican system of the universe.