If, in a large number of drawings, sequences of white (or of black) balls are recorded on occasions, the total number n of drawings will be expressed by n=x1Y1-1-x2Y2-Fx31734-• . . • . . . The relative frequencies of the different series are yl, y2, etc., where n.y1= Y1, n-y2= Y2, etc., and, when n is indefinitely increased, the relation between the x's and the y's is expressed by the formula which represents a curve of the shape shown below in fig. (2), known as the probability curve.
It is not proposed to enter into discussion of the character of the probability curve, or the interpretation of the variations. It will be sufficient here to point out that the equation given relates to a curve symmetrical in form to right and left of the position represented by x= o. The part to the right may be supposed to show the record of white sequences, and that to the left the exactly similar record of black sequences, i.e., of sequences in which white balls failed to appear.
In any actual series of observations (e.g., of the colours of balls drawn from, and returned to, a bag) the observed numbers expressing the frequency of different events will be found to differ more or less from those expressed by the curve, of the form given by the above equation, which is appropriate to the circum stances of the experiment. Prolonged trials would give results approximating to those derived from theoretical calculations and expressed in the shape of the probability curve. Thus, in the case of the heights of 73o men, it is probable that records covering larger numbers of men, of the same race and social condition and within the same limits of age as those from which the 73o were selected, would give a distribution of heights more closely corre sponding to that shown by a typical probability curve than the numbers cited. If, for example, for every one actually covered by our table, io or 20 or 5o or ioo had been measured, closer correspondence with a suitably selected theoretical series would have been probable.
In considering the meaning of changes in the observed fre quency of events, it is generally of considerable importance to ascertain what is the extent of variation that is as likely to happen as not, since the significance of actual variations can only be judged in relation to those which may have no significance at all in reference to the problem under consideration.
forms, and of their theoretical bases, has been particularly active since the last decade of the 19th century. It will suffice here to mention that particular statistical problems are found to yield distributions of observations, not only symmetrical on either side of their mean, as with the normal probability curve, but also grouped more closely on one side of the mean than on the other. The illustration from men's heights showed some small tendency to a skew shape, though it appears possible that the apparent de formation would, on extending the field of observation, be found to disappear and be shown to result from the fact that, in so small a number of cases as 73o, the even representation of men of all heights in the population from which these cases were drawn had not been exactly secured. The following records of weights of men of 25 and less than 3o years of age and within half an inch of 5 ft. 6 in. in height, shows a more marked skewness.
In this case the average (or arithmetic mean) of the weights of the 4,936 men is, on the assumption of continuous distribution, approximately 141.6 lb. The median is at about i lb. less, viz., 14o.5 lb. In the previous illustration of heights, the median was greater than the mean, so that the skewness shown is of opposite direction in the two cases.
This illustration serves also to show a third characteristic of statistical distributions of this kind. Nearly four of every nine cases are included in that one of the eight groups of which 135 lb. is the central point, this group being notably more numerous than any other. The weight 135 lb. may be called, so far as the figures furnish a ready indication, the most generally occurring weight or the "modal" weight. The point in the distribution thus determined is called the "mode." A table like the above does not, however, give a very close measure of the central point of the group (covering a range of 15 lb.) which would prove largest if we compared various groups such as 1271 to 1424 lb. (the group shown) 127 to 142 lb., 128 to 143 lb., and so on. If the curve be found which represents the distribution in question, its highest point is the point the weight corresponding to which is the "mode" of the distribution, the most prevalent weight in the distribution under examination. Various problems in probability lead to curves of the distribution of chances which show the skew ness that marks groups of statistical observations.