In the social sciences, especially in the field of economics, the statistical method is equally indispensable. No economist would attempt to arrive at a conclusion concerning the produc tion or distribution of wealth without an ex haustive study of statistical data. Business men and economists alike are convinced that only when Congress bases its tariff legislation upon a scientific study of the natural resources and economic needs of the various parts of the country will an equitable tariff law be passed. Careful analysis of the effects of the English system of "out-relief," the granting of partial relief to paupers without sending them to the poorhouse, showed that the system was lower ing wages and increasing instead of decreasing pauperism, thus producing a social evil which grew as it fed on itself. In economics and the other social sciences the old Roman proverb is pre-eminently true, "Sine numero nihil dem onstrandum est.° In seeking to apply the statistical method to any line of investigation care must be taken to avoid certain errors likely to be encountered where statistical data are inaccurately compiled or carelessly applied. One of the most common causes of error in conclusions derived from statistical data is the lack of proper definition of the elements entering into the problem con cerning which data are being gathered. Other errors arise from careless or premeditated abuse of data, correct in themselves, but leading to erroneous conclusions when improperly used. False statistical deductions may result from in accuracy, from slovenliness in gathering or ar ranging data, from partial statement of truth, from a comparison of heterogeneous qualities, or from lapses in logic. The man who thought he had discovered that eminent mathematicians were remarkable for their longevity was cor rect in his data, but careless in their applica tion. He should have compared the average length of life of eminent mathematicians, not with the average length of life of all human beings including those who die in infancy, but with the average length of life of men who at tain intellectual maturity. Again, it is mani festly unfair to judge the skill of the physician by the number of his patients who have died, without also considering the total number he treated. Such abuse of statistics as the use of the same set of figures to prove that prohibi tion does prohibit, and that it does not prohibit has, to some extent, destroyed the popular faith in statistics and is responsible for the 'often heard expression, "You can prove anything by figures." Statistics may be likened to a razor, a delicate and very effective tool when handled by an expert but very dangerous to an infant. While care must be taken that accuracy is pre served in the gathering of statistics too great minutia in description defeats the object sought. If, for instance, we wish to study the distribu tion of wage-earners in industrial establish ments we will find ourselves lost unless we effect a grouping into a few classes, as there is a rather small limit to the number of im pressions which a mind can receive and hold at one time. Closely related to this error of excessive attention to minutia of detail is the attempt at greater accuracy in statistical de ductions than the facts themselves warrant. There is in statistics much swallowing of camels and straining at gnats. We are continually confronted with financial statistics carried out to the last cent when many of the individual items cannot be ascertained with any degree of certainty to within the nearest thousand, or even million. The young man who, wishing to know the circumference of a wagon wheel, mul tiplied the diameter, obtained with a 35-inch yard stick, with the value of r carried out to 18 decimals, is illustrative of our ingrained but untrained desire for accuracy. That no resultant can be more accurate than the least accurate of its constituents is an axiom that is altogether too often forgotten, and much time and labor have been spent in efforts to secure precision not warranted by the facts. On the other hand no statistical conclusions should be adopted without a statement indicating the degree of reliance which may be placed on them. It must be borne in mind that statistics is "the science of large numbers," that results ob tained from a few cases cannot be considered representative and should, therefore, be received with due caution. It can readily be proved that the precision of a determination is pro portional to the square root of the number of observations, but also that there is a limit beyond which it does not pay to carry the re finement.
b. Statistical Tables.— In employing the statistical method statistical tables, both simple and complex are constantly brought into use. To illustrate the construction of these we will use the problem, referred to in a previous para graph, of the distribution of wage-earners in the United States. The 1910 census (Vol. VIII) divides the industrial establishments of this country into the following seven classes: those employing 1 to 5 wage-earners; 6 to 20; 21 to 50; 51 to 100; 101 to 250; 251 to 1,000; and over 1,000. We construct a table with these seven classes in one column and in a second column enter the number of establishments in each class. For comparative study we now enter in a third column the number of establishments in each of these classes as given in the census of 1900. This suggests a fourth column giving the increase or decrease for each class, and still another column giving the proportions, ex pressed as percentages, which such increase or decrease bears to the figures given for the earlier of the two years. Such a table is called a simple statistical table. Similarly we may construct another table, but giving here the number of wage-earners in each class. A combination of the two into a double-column table, such as is given in the census of 1910, is called a complex statistical table. Applying to this the proper arithmetical operations gives numerous interesting and important facts.
c. The Use of the Relative.—While satis tics are absolute facts, a study of them is really a study of relatives. In the above problem this was shown by the introduction of propor tions. In that problem the number of establish ments employing 1 to 5 wage-earners, or the number of employees, means little unless we also know the proportion which these numbers bear to the total. This proportion is, in general, best expressed as a percentage. In vital statis tics, however, it has become customary to use a radix number; so many deaths per thousand, for instance. Except in actuarial mathematics, where the formula have been developed on this basis, the latter is not a good method, A it does not readily yield itself to mathematical treatment.
d. Graphic Methods.—To show vividly the facts compiled in statistical tables and the conclusions to be deduced therefrom the statis tician uses various graphic illustrations. What the microscope is to the biologist and the tele scope to the astronomer, of like importance is this mechanical interpretation or illustration of statistics to the statistician. The common housefly is to the casual observer an uninter esting and repulsive creature, but the micro scope reveals it as an object of wondrous struc ture and beauty. Similarly a table giving the temperature, taken each half hour during a cer tain day, means very little to us, but the curve automatically drawn by a thermograph records the changes in temperature momently and still tells us its whole story at a glance.
To construct such a frequency curve, as it is called, where variation is reasonably continu ous, as in the above problem, erect at equal intervals along a horizontal line a series of vertical lines called ordinates, proportional in length to the number of objects in the classes which they represent. Then join the tops of these ordinates in such a way as to make a smooth curve. Such a curve not only makes clear to us the fluctuations in the frequency of its variates, which in the table are effectively hidden, but it also enables us to make import ant comparisons. We may, for instance, draw, on the same diagram but in different colors or with different types of lines five curves representing the total gold and silver in the national treasury, and the amount of silver, silver certificates and treasury notes in circu lation at the end of each quarter for a number of years. We have thereby before us the whole financial history of the country during this period and are, furthermore, able to make both interesting and valuable deductions by com parisons of these curves. These curves play an important part in the physical and biological sciences as they enable the scientist to replace the rough approximations obtained from a few observations with the law gpverning the frequency distribution. As aids in inter polation and extrapolation these curves are in dispensable. This branch of statistical inquiry has been developed to its present high stage of perfection by Prof. Karl Pearson, who, through his work on the 'Mathematical Theory of Evo lution,) has breathed into the clay of dry statis tical data a living soul.