MEAN. By the mean of two or more quantities is meant an inter mediate quantity determined by mathematical rules. There are more ways than one of finding a mean, but the two principal results of this kind are called the arithmetical and the geometrical means. The names are not properly expressive of the distinction between them, but they are established by use.
An arithmetical mean is the simple AVF.11A0t, formed by adding the quantities together, and dividing by the number of quantities. A geometrical mean is the square root of the product of the quantities. I knerally, let there be a number of quantities, x„ x„, .r,, fin., and let (.r , x„ x„ &c.) be a function of them which is symmetrical, that is, which Is not altered when*any two of them are interchanged ; then if y be found from the equation (Y, y, &Om 4' (xi, &c.), y may be called a species of mean.
The arithmetical mean, or average (which is always to be understood when the word mean Is mentioned, unless the contrary be specified), is taken to be the must probable result of a number of discordant quantities, which would have been the same but for errors of obser vation or experiment. Thus if three measures of the same length give 122, 123, and 123.4, the mean of which is it is presumed that 122'8 is more likely to be the real length which was attempted to be measured than any other. We confine ourselves in the present article to pointing out how it may be ascertained what degree of probability belongs to such results.
In assuming the average as the most probable result, it is presumed that any one measurement is as likely to err one way as the other ; that is, as likely to be too small as too great. If nothing but results be known, this presumption is justifiable ; but if it be known that there is more tendency to error of one sort than the other, the most pro bable result cannot be ascertained until it is found out by how much the average of a very large number of observations would be affected by this tendency. Say it is known that in the long run the average will be increased .3 above the truth by a greater tendency to measure too much than too little; then 122.8—.3, or is the most pro bable result of the preceding three observations.
It is obvious that when observations nearly agree with each other, the average must be nearly the truth required, and the nearer the agreement of the observations, the more nearly. If the observations do not agree well, the average is still more likely than anything else, but not so likely as before.
We nop show how, having a number of observations, to determine the probability that the truth lies within a given degree of nearness to the average. A table must be used, which we here give to a greater extent than we should otherwise do, on account of succeeding articles. Let am be time average of a number of observations, and let al +sa and at—ta be the limits of which it is required to know what is the chance of the truth being between them. Take the difference between al and each of the results of observation, and add the squares of these dif ferences. Multiply 100 times the number of observations by 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 fur the degree of nearness required.
Suppose, for example, that seven observations give 10'03, 10.71, 10'26, 10'30, 1012, 10'81, the average of which is differing from the respective observations by '51, 17, '44, '28, .24, 18, and '27, the sum of the squares of which is 1239, twice which is I•4478, the square root of which is Let it be required to find the chance of the truth lying between and we have then to multiply 700 by .06, which Fives 42, and to divide by 1.203, which gives 34.9. Opposite to 35 in the column A is found 3794, so that 3794 out of 10,000, or 3794 to 6200 is time chance of the result lying between the limits given : that is, nearly 31 to Ifo against it. If the limits, proposed had been 10'54+1 and 10'54-1, 700 multiplied by 1 and divided by I•203 would have given 58.2. Opposite to 58 in the table we find 5879, so that it is 5879 to 4121, ur about 59 to 4], in favour of the result lying between and 10:44.