In the preceding rule it is supposed that all the observations arc equally trustworthy, or that there is no circumstance which would beforehand lead us to suppose that any one is more likely to be true than another. If this be not the case, no rule can be applied except one which depends on the observer's judgment. He must make the different observations reckon as different nr nbers of observations, allowing any one observation to count as mon, than one, if he believes it to be better than the rest. Thus, suppose three observations to give 26, 28, and 29, and that it is thought there is reason to prefer 28 to the others, and 29 to 26, so that 28 ranks in the observer's mind as being as good as a mean of eight observations, 29 of six, and 26 of four. It must then be considered that there have been 8 + 6 + 4, or 18 obser vations, of which 8 have given 28, 6 have given 29, and 4 have given 26. These numbers, 8, 6, and 4, are called the weights of the several observations 28, 29, and 26, and the alteration in the preceding rule is as follows :-In forming the average, multiply each observation by its weight ; add the result, and divide by the sum of the weights. Thus 8 x 28 +6 x 29 + 4 x 26=502, which divided by S +6 +4, or 18, is 27.89, the most probable result. In finding the probability of the truth lying within given limits on one side or the other of this most probable average, let the average be m as before, and the limits 31+ no and al-m; take the difference between at and each of the results of observation, multiply the square of each difference by the weight of its observation, and add the results. Multiply 100 times the sum of the weights by ra, and divide by the square root of twice the sum just found ; take the number nearest to the result in the column marked A, and opposite to it in the column marked B will be found the number of chances out of 10,000 for the degree of nearness required. Thus if in the preceding instance we ask what is the chance of the truth lying between 27'89 + .2 and 27'89-.2, we observe that 27.89 differs from the several results by 1'89, '11, and 1'11, the squares of which multiplied by 4, 8, and 6, and the results added together, give 211778, twice which is 435556, the square root of which is And 100 times the sum of the weights, or 1800, multiplied by .2, is 360, which divided by 6.6 gives 54.6. Opposite to 55 in column A we find 5633 in column B; that is, we have 5633 to 4367, or about 56 to 44 in favour of the truth lying between the limits specified.
The inverse problem is as follows : given the observations, required the limits of difference from the average between which it is a given chance, a to b, that the truth shall lie. In both cases the first process ie to turn a : (a -1-1) into a decimal fraction of four places, and to take the numerator of such fraction.
Look for the numerator in column B, find the number nearest to it, and take out the number corresponding in column A. Multiply this by the square root used in the direct rule, and divide by 100 times the number of observations, or, if they are not equally good, by 100 times the sum of the weights. The quotient is the answer required. But when, as most frequently happens, an even chance is the given chance, use 471 instead of the number found in column A.
In the first of the given instances it is required to know within what limits it is 99 to 1 that the truth is contained. Here 99 : (99 + 1) is and looking for 9900 in column B we find 9899 opposite to 182 in column A. Multiply 182 by 1.203, which gives 218.946, which divided by 700 gives '313, so that it is 99 to 1 that the truth lies between and In the second instance, required the limits within which it is an even chance that the truth is contained. Multiply 471 by and divide by 1800, which gives '175; and it is an even chance that the truth lies between and The amount of departure from the average within which, on one side or the other, it is an even chance that the truth shall lie, is called the probable error of the observation or average of observations to which it refers. When the probable error of any one observation is given, that of the average is found by dividing it by the square root of the number of observations. Thus, if there be 100 observations, of each of which it is an even chance that it is within .1 of the truth ; then the square root of 100 being 10, and '14-10 being -01, it is an even chance that the average of the hundred observations is within of the truth.
For further account of the matters contained in this article, see PROBABILITIES, THEORY Or; OBSERVATION; RISK. For description of methods without demonstration, see Lardncr's Cabinet Cyclopadia, Essay on Probabilities, or Poisson on Probabilities. The tables may be found to greater extent in the first article cited, and also in the Berlin Astronomisches Jahrbuch for 1834.