The reader will remark that the correspondence is very close, except as regards larger errors, which are excessive in practice. It is one objection, indeed, to the theory of error, that, being expressed in a continuous mathematical function, it contemplates the existence of errors of every magnitude, such as could not practically occur; yet in this case the theory seems to under-estimate the number of large errors.
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:—9
| Magnitude of error in tenths of a second. | Number of Errors | Magnitude of error in tenths of a second. | Number of Errors | ||
|---|---|---|---|---|---|
| by Observation. | by Theory. | by Observation. | by Theory. | ||
| 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;