WEIGHT OF OBSERVATIONS. This article, is only for the reference of the mathematical student; in MEAN will be found as much of it as an arithmetician can use by rule.
This term was first applied in the manner stated In the article MEAN. An observer decided the relative merit of his observations by his unassisted recollection of the impression made by them upon his mind at the time, and affixed weights to them ; that is, supposing A,„ &c,, to be the a results of observation, he attached numbers c„ &c., pro portional to their presumed goodness, and used instead of ro+.9a, for the average. Instead of c„ &e., any numbers propor tional to them may obviously be used : and iu applying the higher branches of the theory of probabilities, it was found that a certain mode of obtaining c„ &e., while it gave the above mode of using these numbers in the formation of an average, made them applicable to other important wee. We here give a sketch of the results of this method in its simplest parts.
1. When a number of discordant observatians, made under circum stances in which positive and negative errors are equally likely, do not differ much from each other, and when it is exceedingly unlikely that the truth can differ much from the observations, it may be presumed that the chances of the error .of any one of those observations lying between x and x d x, and between a and b, are severally of the forms where c is a constant dependent on the goodness of the observations, and r= 3'1l139..., s=2.71828...., as usuaL Even if this law of error do not exist, it is found that the treatment of a considerable number of observations, whatever* may be the law, is reducihle to the same rules as those derived from this law, which is now universally assumed by those observers who apply the theory of probabilities to their results.
2. The constant c is called the weight of the observations, and depends upon the various circumstances which determine their good ness or badness. The greater it is the better the class of observa tions to which it applies. It is approximately found, for a given class of observations, as follows :—Subtract each of the observations from their mean, and let e„ &c., be the results; then cs--n-s-2 The sum of the squares of the departures from the average may be found by diminishing the sum of the squares of the observations by n times the square of the mean; and before doing this any convenient quantity may be struck off from all the observations, provided it be also struck off from the mean.
8. The probable error is that within which, taken positively and negatively, there is an even chance an observation shall lie. Thus if there be an even chance (A being the true result) for the result of an observation lying between A—a and A + a, then a is the probable error of an observation. To find the probable error, divide -476936 by the square root of the weight.
4. The weight of the average of observations is tho sum of tho weights of the component observations. If e observations, „ a„ &c . be made, all of the same weight r, the average is 1.asers, the weight of the average is se, and its probable error is .476936+ V(se). But if the weights be different, say e„ etc., then Ica-e-le is the average, lc is its weight., and .476936+ V(7.c) its probable error. In the former we the probable error of the average may be directly found from the sum of the squares of the reputed errors," by the formula .67449 V(Ie")+ 5. Catcris paribus, the probable error of an average will not be in. Tersely as the number of observations, but as the square root of that number. If p be the probable error of an observation, and r that of the average of n such observations, then p. Vs . P. An observer who takes such a mode as gives the probable error of an observation twice as great as it need be, must not hope to indemnify himself for his carelessness by making twice as many observations as would otherwise be necessary, but must make four times as many.
6. If p be the probable error of an observation, an average, or other result, the following table will be sufficient to connect the probable error with other errors, for any rough purpose of estimation :— This table is to be interpreted as follows :—If p be the probable error above mentioned, it is 14 to 1, or 3 to 2, against the error turning out less than xp, and it is lb to 1 for the error turning out less than P25 x p. It ie 8 to 1 against the error being leas than .21 x p, and 8 to 1 for its being less than 2.36 x p. It im 1000 to 1 against the error being less than .002 x p, and 1000 to 1 for the error being leas than 4.90 x p.