Let us call the four series of inaccurate measures obtained with the four errors, Xa, Xb, Xc, Xd, and Ya, Yb, Yc, Yd.

Call the series obtained by averaging each member of Xa with its correspondent in Xb, Xab.

Let Xcd, Yab and Ycd have similar meanings.

Call the series obtained by averaging each member of Xa with the corresponding Xb, Xc and Xd, Xabcd.

Let Yabcd have a similar meaning.

We have then 4 very inaccurate measures of X in every one of the 100 pairs; so also of Y. We have two less inaccurate measures of X and also of Y in each pair. We have one still better measure, the best obtainable from our data.

We may then calculate the corrected r according to Spearman, using many different combinations of the r's obtained from the above series. The combinations which I have used and the results follow in Table XVI.1

The correspondence of the coefficients corrected by Spearman's formulas with the actual coefficient from accurate measures is satisfactory.

$9. Minor Advice to Students of Mental and Social Relationships As a rule nothing should be taken for granted about any relationship and the result of any calculation should be to express, not to replace, the comprehension of a fact about the series of individual relationships.

'In all the calculations I have assumed the original 0 as the central tendency from which to reckon deviation values. To have turned each of the 200 values of each of the fourteen series into a new deviation measure would have added practically nothing to the general result in the way of accuracy. The labor of 2,800 little sums in addition and 2,800 copyings of numbers could be more profitably spent. My figures are on this basis.

Measurements should be on the finest scale that can be recorded without special difficulty. The attenuation by chance error is thus diminished and the time taken in making a more elaborate correlation table can be saved ten times over by the use of the Median Ratio.

The central tendency from which one measures deviations should be chosen with care so that it stands for some reality divergence from which is significant.

In the relationship given in Table XVII., for instance, from what point should one reckon deviations? The authors take the mean, 56.568. But there is much to be said for taking the modal adult life (at about 70), since that represents an important real tendency and the force of heredity in determining departures from that tendency is perhaps more important than its form in determining departures from the rather arbitrary age, 56.568. The re

of 49 43 47 51 52 55 57 59 62 65 67 68 65 73 74 76

Arrays

Pearson Coefficient .2853, C.T. being 56.568.

Median Ratio = .479, C.T. being 59.4.

1 The relationship between brother and brother in length of life in cases where both brothers are 21 or over, from 'The Inheritance of the Duration of Life' by M. Beeton and K. Pearson, Biometrika, Vol. I., p. 84. I have divided the array of 58 so as to make a median sectioning of the series. In the original the array for 58 is given simply as 14, 7, 12, 6, 8, 15, 10, 12, 11, 21, 8, 19, 11, 3, 2. I have also added approximate medians of arrays.

lationship in the latter case (70.5 being taken as the central tendency) is closer, the Median Ratio being .54 or about 7 higher than the Median Ratio when divergences are calculated from 56.5. The Modal Ratio is unchanged.

It should be evident from the facts stated in previous sections that it is out-and-out folly to be content with calculating for every relationship studied the same type of coefficient. Nothing short of the entire correlation table is the adequate measure of the relationship in question. Any measure of one central tendency of relationship may be misleading, for the relationship may be bimodal. When the observed modal relationship is clearly not near the Pearson Coefficient the latter should be accompanied by the former. So also if the modal relationship is clearly not near the median relationship.

The averages or medians or modes of the arrays should be cal

culated and stated, and unless the relationship is uniform (within the limits of chance error) throughout the course of the series a most probable curved line to fit the entire series should be calculated instead of the slope of a straight line.

For instance, the Pearson Coefficient for the relation between adult brother and adult brother in longevity is given by Beeton and Pearson as .2853. The relation is sufficiently close to uniformity for all values of x to make a linear relation at least approximately true (if we consider also the similar relation between sister and sister). The relation is, however, by no means identical with other relations giving a similar coefficient, for the modal relationship is approximately 1.00. This can be seen at a glance from the graphic representation of the correlation table (Fig. 10) or the distribution of the ratios (deviations are reckoned from 59.4 and 59.4 as central tendencies) in Fig. 11.

The 2853 then does not mean that the most likely value of B — C.T. of B is near .2853 × (A — C.T. of A), nor that the forces producing correlation tend to make B/A=.2853, divergencies from

15-40 40-65 65-90 90-115 115-140

FIG. 11. Frequencies of different degrees of relationship in the case of fraternal longevity. The numbers stand for the ratios in per cents, the heights for their relative frequency. The mode is at very, very close resemblance, or ratios of 90 to 115 per cent.

this being due to minor causes producing variations in the correlation. On the contrary the .2853 represents a most ambiguous summation of the force of a tendency to identical longevity and many other forces. If the authors had not given the full correlation table, the .2853 would evidently have been definitely misleading.

The determination of the most likely law of relationship for a series of pairs may then be theoretically and practically a different problem for each particular case, a problem to solve which we need not only certain mathematical technique but also abundant knowledge of other similar relationships and of the entire body of facts relevant to the relationship in question. Thus the same set of pairs could properly be interpreted on the basis of a linear relationship

when they were male brothers' first-rib lengths, and could not properly be so interpreted if they were related body-strengths and earnings in dollars. For we have evidence from cephalic index, stature and the like to justify some expectation of linear correlation for fraternal relationships in features of anatomy, whereas what evidence we have concerning the relationship between body-strength and earning capacity in individuals goes to show that it is far less close for those of high earning capacity than for those of very low earning capacity.