780 PROBABILITY of its correctness without proof. Let the lists of two of the judges be, beginning from the lowest, B , II , R . , K , A . . . . C , K , D , H , B . . . . Probabilities A : , A., , A 3 , 4 , X 5 . . . . As the opinions of all the judges are supposed of equal weight, the cause H nere is as likely as the cause K ; but the probability that H or K was the cause is 1 ., + 4 . Hence prob. (H) + prob. (K) = 2 prob. (II) = A, + A 4 ; .-. prob. (II)-i(Aj + A 4 ); that is, the probability of any cause is the mean of its probabili ties on the two lists, the circumstance being clearly immaterial whether the same cause K is found opposite to it or not. The same follows for 3 or more lists. 43. Laplace applies the same method to elections. Suppose there are n candidates for an office ; each elector is to arrange them ia what he believes to be the order of merit ; and we have iirst to rind the numerical value of the merit he thus implicitly attributes to each candidate. Fixing on some limit a as the maximum of merit, n arbitrary values less than a are taken and then arranged in order of magnitude least, second, third, .... greatest; to find the mean value of each. till i A X Y Z B Take a line AB = a, and set off n arbitrary lengths AX, AY, AZ .... beginning at A ; that is, n points are taken at random in AB. Now the mean values of AX, XY, YZ, .... are all equal ; for if a new point P be taken at random, it is equally likely to be 1st, 2d, 3d, &c., in order beginning from A, because out of n + l points the chance of an assigned one being 1st is (n+l) 1 ; of its being 2d (71 + 1)- 1 ; and so on. But the chance of P being 1st is equal to the mean value of AX divided by AB ; of its being 2d M(XY)4-AB; and so on. Hence the mean value of AX is AB (?i + 1)- 1 ; that of AY is 2AB(?i + l)- 1 ; and so on. Thus the mean merit assigned to the several candidates is aOi + 1)- 1 , 2a(i + l)- 1 , 3(/i + l)- 1 . . . . 7ia(n + l)-i. Thus the relative merits may be estimated by writing under the names of the candidates the numbers 1, 2/3, . . . . n. The same being done by each elector, the probability will be in favour of the candidate who has the greatest sum. Practically it is to be feared that this plan would not succeed, though certainly the most rational and logical one if the conditions are fulfilled because, as Laplace observes, not only are electors swayed by many considerations independent of the merit of the candidates, but they would often place low down in their list any candidate whom they judged a formidable competitor to the one they preferred, thus giving an unfair advantage to candidates of mediocre merit. Tkere are, however, many cases where snch objections would not apply, and therefore where Laplace s method would be certainly the most rational. Thus, suppose a jury or committee or board of examiners have to decide on the relative merit of a number of prize essays, designs for a building, &c. ; each member should place them in what he judges to be the order of merit, beginning with the worst, and write over them the numbers 1, 2, 3, 4, &c. ; then the relative merit of each essay, &c., would be represented by the sum of the numbers against it in each list. No doubt there would be cases where a juror would observe a great difference in merit between one essay and the one below it, which difference would not be adequately rendered by an excess of 1 in the number. But even then, as such superiority could not fail to be recognized by the other members of the tribunal, it is not likely that any injustice would result. 44. An argument advanced in support of a proposition differs from the case of testimony in that, if the argument is bad, the previous probability of the" conclusion is unaffected. Let p be the 11 priori probability of the proposition, q the chance that the argument is correct ; then, in a large number N of cases, in <?N the argument is good, and therefore the proposition is true ; and out of the remaining (1 -q) N, where the argument is bad, there are p (1 -?)N cases where the proposition is nevertheless true. Hence the probability of the conclusion is p+q-pq. Hence any argument, however weak, adds something to the force of preceding arguments. V. Ox MEAN VALVES AXD THE THEORY OF ERRORS. 45. The idea of a mean or average among many differing magnitudes of the same kind is one continually employed, and of great value. It gives us in one result, easily pictured to the mind and easily remembered, a general idea of a number of quantities which perhaps we have never seen or observed, and we can thus convey the same idea to others, without giving a long list of the quantities themselves. AVe could scarcely form any clear concep tion as to the duration of human life, unless by taking the average, that is, finding the length of life each individual would have if the whole sum of the years attained by each were equally divided among the entire population. How, again, could we so easily form a idea of the climate of Rome or Nice as by learning the mean of the temperatures of each day for a year, or a series of years ? Here, again, it will be an important addition to the information to find also the mean summer temperature and the mean in winter, as we thus learn what extremes of heat and cold are to be expected. We may even go further and inquire the diurnal variation in the temperature in summer or in winter ; and for this we should know the average of a number of particular cases. It may be said that the whole value of statistics depends on the doctrine of averages. The price of wheat and of other commodities, the increase or decrease of a particular crime, the age of marriages both for men and women, the amount of rain at a given locality, the advance of education, the distribution of wealth, the spread of disease, and numberless other subjects for inquiry are instances where we often see hasty and misleading conclusions drawn from one or two particular cases which happen to make an impression, but where the philosophical method bids us to observe the results in a large number, and then to present them as summed up and represented by the average or mean. 46. There is another application of averages of a different nature from the foregoing. Different estimates of the same thing are given by several independent authorities : thus the precise moment of an earthquake is differently stated by correspondents in the papers ; different heights are given for a mountain by travellers ; or suppose I have myself measured the height of a building a number of times, never obtaining exactly the same result. In all such cases (if we have no reason to attach greater weight to one result than to another) our common sense tells us that the average of all the estimates is more likely to be the truth than any other value. In these cases, as M. Quetelet remarks, there is this important distinction from the preceding, that the mean value represents a thing actually existing ; whereas in the others it merely serves to give a kind of general idea of a number of indi viduals essentially different, though of the same kind. Thus if I take the mean of the heights of 200 houses in a long street, it does not stand for any real entity, but is a mere ideal height, repre senting as nearly as possible those of the individual houses, whereas, in taking 200 measurements of the same house, their mean is intended to give, and will very nearly give, the actual height of that house. 47. So far it is obvious how to proceed in such cases ; but it becomes a most important question in the theory of probabilities, to determine how far we can rely on the mean value of the different observations giving us the true magnitude we seek, or rather, as we never can expect it to give exactly that value, to ascertain with what probability we may expect the error not to exceed any assigned limit. Such is the inquiry on which we are about to enter. This investigation is of the more importance, because we find what is really the same problem present itself again under circum stances different from what we have been considering. In the measurement of any whole by means of repeated partial measure ments as, for instance, in measuring a distance by means of a chain the error in the result is the sum of all the partial errors (with their proper signs) incurred at each successive application of the chain. If we would know, then, the amount of confidence we may have in the accuracy of the result, we must determine, as well as we can, the probability of the error that is, the sum of all the partial errors not exceeding assigned limits ; and to this end, we have in the first place to try to determine the law of facility, or frequency, of different values of this sum. The problem only differs from the preceding in that here we seek for the facility of the sum of the errors ; in the farmer, of the nth part of that sum. In both these cases, we may reasonably and naturally suppose that the error incurred in each observation, or each measurement, follows the same law as to the frequency of its different possible values and as to its limits, as each is made by the same observer, under the same circumstances, though what that law is may be unknown to us. But there is another class of cases where the same problem presents itself. An astronomical observation is made (say) of the zenith distance of a star at a particular instant; the error in this determination is a complex one, caused by an
error in the time, an error in the refraction, errors of the instru-