The Theory of Errors

This is directly connected with the problem of the probability of causes.

Here again we find effects—to wit, a certain number of irreconcilable observations.

We try to find the causes which are, on the one hand, the true value of the quantity to be measured, and, on the other, the error made in each isolated observation.

We must calculate the probable à posteriori value of each error, and therefore the probable value of the quantity to be measured.

But we cannot undertake this calculation unless we admit à priori —i.e., before any observations are made—that there is a law of the probability of errors. Is there a law of errors? The law to which all calculators assent is Gauss’s law, that is represented by a certain transcendental curve known as the “bell.”

But it is first of all necessary to recall the classic distinction between systematic and accidental errors. If the metre with which we measure a length is too long, the number we get will be too small, and it will be no use to measure several times—that is a systematic error.

If we measure with an accurate metre, we may make a mistake, and find the length sometimes too large and sometimes too small, and when we take the mean of a large number of measurements, the error will tend to grow small. These are accidental errors.

It is clear that systematic errors do not satisfy Gauss’s law, but do accidental errors satisfy it?

Numerous proofs have been attempted, almost all of them crude paralogisms.

But starting from the following hypotheses we may prove Gauss’s law: the error is the result of a very large number of partial and independent errors; each partial error is very small and obeys any law of probability whatever, provided the probability of a positive error is the same as that of an equal negative error.

These conditions will be often, but not always, fulfilled, and we may reserve the name of accidental for errors which satisfy them.

We see that the method of least squares is not legitimate in every case; in general, physicists are more distrustful of it than astronomers.

Both physicists and astronomers suffer from systematic errors. The astronomers have to contend with an extremely important source of error which is entirely accidental—I mean atmospheric undulations.

So it is very curious to hear a discussion between a physicist and an astronomer about a method of observation.

The physicist is persuaded that one good measurement is worth more than many bad ones. He is pre-eminently concerned with the elimination by means of every precaution of the final systematic errors.

The astronomer retorts: “But you can only observe a small number of stars, and accidental errors will not disappear.”

What conclusion must we draw? Must we continue to use the method of least squares? We must distinguish.

We have eliminated all the systematic errors of which we have any suspicion; we are quite certain that there are others still, but we cannot detect them; and yet we must make up our minds and adopt a definitive value which will be regarded as the probable value; and for that purpose it is clear that the best thing we can do is to apply Gauss’s law. We have only applied a practical rule referring to subjective probability.

Yet we want to go farther and say that not only the probable value is so much, but that the probable error in the result is so much.

This is absolutely invalid : it would be true only if we were sure that all the systematic errors were eliminated, and of that we know absolutely nothing.

We have two series of observations; by applying the law of least squares we find that the probable error in the first series is twice as small as in the second.

The second series may, however, be more accurate than the first, because the first is perhaps affected by a large systematic error. All that we can say is, that the first series is probably better than the second because its accidental error is smaller, and that we have no reason for affirming that the systematic error is greater for one of the series than for the other, our ignorance on this point being absolute.

7. Conclusions

To undertake the calculation of any probability, and even for that calculation to have any meaning at all, we must admit, as a point of departure, an hypothesis or convention which has always something arbitrary about it. In the choice of this convention we can be guided only by the principle of suffi- cient reason. Unfortunately, this principle is very vague and very elastic, and in the cursory examination we have just made we have seen it assume different forms.

The form under which we meet it most often is the belief in continuity, a belief which it would be difficult to justify by apodeictic reasoning, but without which all science would be impossible. Finally, the problems to which the calculus of probabilities may be applied with profit are those in which the result is independent of the hypothesis made at the outset, provided only that this hypothesis satisfies the condition of continuity.