18. All of these facts establish a strong probability that the original samples were well chosen-a fact that the authors of the

018=$2 +5

articles from which the data were drawn did not take pains to establish to the satisfaction of the critical reader. This statistical See No. 7. Several years before these Chicago studies, Miss M. F. Byington successfully used at Homestead an empirical method that approximated closely to scientific requirements.

FIG. 3

analysis is, therefore, valuable as an example of how it is possible to check back on the fieldwork procedure of an investigation in cases in which the description of the investigation supplied by the surveyors is faulty in this respect.

19. It may be worth while at this point to make an observation which has some value in theoretical statistics as well as significance for practical investigation. It was stated in paragraph 5 that the standard error of the mean could be used as a device to test the reliability of the average of a sample, whether or no the distribution of items in the universe or the sample was skew. The reason for this is, briefly: (1) when random samples are selected, the mean of each sample becomes as it were a measurement on the true mean of the universe from which the sample was selected; (2) the true mean of the universe thus becomes in a sense a constant value which we attempt to approximate; (3) the measurements on a constant if numerous enough tend to obey the law of error. Now, the law of error is that in measurements upon a constant there is a welldefined tendency when measurements are numerous: for small errors to be more frequent than large errors, for positive and negative errors to be equally frequent, and for large errors to occur infrequently if at all. Now, the analysis just concluded, in paragraphs 16-18 inclusive, shows that even in our study of eleven means there is a well-defined tendency for the means of the samples to approximate this law of error, and so we have in this study a very neat example of the application of this important mathematical-statistical principle.

20. One more statistical test of sampling remains to be applied. When two samples are picked from a universe and we find that there is a difference between the values of their respective means, we may ask the question: Is the difference significant of real differences in the universe such that there may really be two universes when we had thought that there was only one? Or, on the other hand, may the difference be due merely to the fluctuations of simple sampling? The statistical rule is that when the difference between two means is less than three times the standard error of the means, we may conclude that the difference is not significant. The I Chapin, op. cit., and A. L. Bowley, Elements of Statistics (1901 ed.), pp. 303, 308. Yule, op. cit., pp. 345-46.

formula for computing the standard error of the difference between two means is as follows:

Table V gives the errors of differences between the means of the first eight samples; for example, Nos. 9 and 10 are clearly outside the allowable range, and, consequently, have not been computed. It will be observed that differences between the means of samples

2, 3, 4, 5, 6, and 7 are not significant, but that when the means of samples 1 and 8 are introduced, we immediately get a significant difference. This table is, therefore, an additional check on the conclusions given in paragraphs 18 and 19.

21. Before concluding this treatment it is important to consider the meaning of columns 9 and 10 of Table IV. The series of coefficients of correlation presented in column 9 are surprisingly uniform. This means that conditions of room overcrowding are not substantially different among the eleven samples, the correlation is low in almost every case, and in most instances it is a negative correlation. This develops the surprising point that as the number of occupants of a room increases there is a slight tendency for the size of the room to diminish. At any rate, the relationship, whether

positive or negative, is not marked. The meaning of column io, "The Probable Errors of the Coefficients of Correlation," is explained in Figure 4. The vertical lines represent the probable error range for each of the eleven coefficients of correlation; for example, in sample 3, r= -.148±.031, which means that, judging the correlation from this sample alone, the chances are even that the true correlation lies between r=.117 and r = -.179. Similarly, for the other coefficients. It will be observed at once that the coefficients derived from samples 1, 2, 3, 4, 6, 7, and II tend to cluster. This would suggest that the true correlation lies between -.10 and -.20. The coefficient of sample 5 because erratic is omitted.

III. SUMMARY AND CONCLUSION

22. We may now make the following points in review of our statistical analysis: (1) The societal variable, overcrowding in sleeping-rooms, is susceptible of quantitative definition in terms of mean, standard deviation, standard error, probable error, coefficient of correlation, skewness, and curve-fitting. By these devices we can discover the law of distribution of the data, and the chief trends or tendencies may be defined with clearer precision than can be true of any verbal description. (2) Applying this type of analysis to eleven samples of overcrowded sleeping-rooms, we have obtained a precise quantitative definition of a societal variable. This statistical comparison and analysis makes it possible to check back on the validity of the original fieldwork. In this case, the essential fairness of the original study seems to be established statistically. (3) It is sometimes said that the statistical method is inapplicable to the study of societal variables. What is meant is that use of refined statistical tools is not justified in the interpretation of data in which the original errors of observation play such a large rôle as in social studies. It would seem that this paper is a refutation of this argument with respect to the subject of room overcrowding at least, since the application of refined statistical methods has established the essential fairness of the original fieldwork-a consideration by no means established in the original articles.