Where variation is not continuous or where discrete objects are under consideration the histogram or • pictogram are preferable. The histogram is merely a succession of rectangles obtained by drawing through the tops of the frequency ordinates. lines parallel to the base. A common and effective pictogram is a circle divided into sectors, the areas of the sectors being proportional to the magnitude of the re spective objects under comparison. For in stance, we can illustrate the relative values of the principal crops of Nebraska by dividing a circle into five sectors of the proper relative sizes and label them ?*corti,* "(oats,* and Gall other drops.* Or we can place, side by side, on equal s bases, five rectangles whose altitudes are; proportional to the values of these crops. A complex table can be illus trated by superimpbsing two such diagrams on each other, as /shown in the preceding figure. The nuinber of 'ays in which such illustrations can be made-is limited only by the limits of human ingentfity.
e. The of the of the important part which they play in statistics, the statistical method has been defined by some writers as Sciente of Averages.* Aver ages may be considered as types through which we get an adequate ,unucptao.i of groups of variates and by means of which comparisons with other groups spay be effected. Thus, though men of any rice may differ greatly in height, we still get a very fair conception of the relative stature of a Patagonian an Englishman, and a Lapp when we learn that their average heights are six feet one inch, five feet seven inches, and five feet, respectively. The first real development of scientific methods in statistical study came about through the needs of astronomers for some means of secur ing accurate determinations from their ever growing stock of observational data It was found errors of observation were symmetrically grouped about their arithmetic mean, where they were clustered most closely, and decreas ing continuously and equally in number in ac cordance with a certain law, called the normal law of errors, as the observation re ceded in either direction from the mean.f This law, also known as the Law of Probability or the Law of Chance, was found by the great Belgian astronomer and mathmatician, Jaques Quetelet, to' govern not merely the distribution of errors of measurement of inanimate things, or direct the action of *blind chance,* such as the marmer in which coins, when tossed, turn up or *tails,* but also the chance dis tributions in the animate world. Whether dealing with barometric pressure, human height or the size of plants the law was found to hold. Later it was found that where a bias
exists, as, for instance, the inclination of the loaded coin to turn. up one face more fre quently than the ..ether, or wages to taper off more rapidly-VS the lower limit, the law still helck, istif in a _generalized form. The place of concentration which is no longer the mean has been called by Professor Pearson the *mode?) There are three forms of averages in mon use, two of which have already been re ferred to. The three are the arithmetic mean, the median and the mode, the first being by far the most widely used. To these may be added the geometric and harmonic means, which, though seldom used, are indispensable in cer tain lines of investigation. The nature of the mode has already been described in the pre ceding paragraph. The median, meaning the halfway point, is especially useful in dealing with non-measurable quantities, such as eco nomic unrest, eye color or mental character istics. We cannot for instance, state in figures the mental alertness of a class of boys, but we can readily arrange them in the order of their intelligence and thus determine the average boy. In a normal distribution the mean, the median and the mode naturally coincide. The arithmetic mean may be subdivided into the simple and the weighted. The simple arith metic mean of two or more quantities is the sum of those divided by their number, with out regard to their respective weights or rela tive importance. The weighted mean is the sum of the quantities, each multiplied by its weight, divided by the sum of the weights. The difference can best be pointed out by some examples. If we wish to find the average wage in an industry we cannot do this by simply averaging the scales of wages paid in the vari ous occupations. If that were done the few, highly-paid experts and superintendents would count as much as the many common laborers and so give a very misleading conception as to the wages really paid. The true average is obtained by weighting each wage by the num ber of men receiving that wage, adding the products and dividing by the number of men. In this case the weights are definitely determin able numbers, which is not always the case. An astronomer taking a series of observations on a certain evening will average these as equally good. When, however, he wishes to combine observations taken on different even ings he may consider these sets as of unequal value because of the unequal atmospheric con ditions under which the observations were made. He will then use as weights numbers which, in his estimation, give the relative val ues of the observations.