Putting together all these differences between Atrue and Btrue, we The chances are 1 to 1 that the true difference will not vary from the obtained difference by more than 1.12, will not go outside of -6.88 and -9.12. The variability of the divergence between the true measures is thus dependent on the variabilities of the divergences of each one from its corresponding obtained measure. The unreliability of a difference between two measures equals in fact the square root of the sum of the squares of the unreliabilities of the measures themselves. The formula for its calculation is, Variability of (At. — Bt.) =√ [var. of (4. — A obt.)]2+ [var. of (Bt. — Bobt.)]2. Using the common standards of measurement of variability, Gdiff. At. - Bt. = √(@At. - Aobt.)2 + (@Bt. - Bobt.)2 =√(A.D.At.-Aobt.)2 + (A.D.Bt.-Bobt.)2 E.At. The most probable true difference is, then, the obtained difference, and the chances that the true difference is so much less or so much more than it can be calculated from the tables for the probability surface. The Reliability of a Pearson Coefficient of Correlation. The divergence, in a case of lineal correlation, of the true coefficient of correlation from that obtained from the limited number of pairs of measures compared, is a variable trait with a probable mode at 0, and a variability which serves as the measure of the unreliability of the obtained result. The formulæ * are: It is customary to speak of the variability of the divergence of true from obtained measure as the measure's error. Thus бt. av.- obt. av. is called the mean square error of the obtained average; P. E..r.-obt. r. is called the probable error of the obtained coefficient of correlation; A. D.t. diff.— obt. diff. is called the average error of the obtained difference. These terms are somewhat ill chosen, as there is really no 'error,' but only a varying degree of probable approximation. I have, therefore, used the word unreliability throughout. PROBLEMS. What is the unreliability of each of the averages and variabilities in the following cases? * There is some uncertainty about these formulæ, certain authorities favoring the use of simply Vn in the denominators in place of √n(1+r2). What is the unreliability of each of the following differences? 35. Av.c- Av. 2. The data concerning Av., and Av.c being as in 30 and 32. = = 36. Av. Av. 3. The data concerning Av., and Av.c being as in 30 and 33. A = 37. Av. Av. 4. The data concerning Av., and Av.c being as in 34 and 30. = 38. Av. Av. 4. The data concerning Av., and Av.c being as in 34 and 31. = 39. Av.E- Av. 2. The data concerning Av., and Av.c being as in 34 and 32. What is the unreliability of r in each of the following cases? CHAPTER XI. THE USE OF TABLES OF FREQUENCY OF THE PROBABILITY SURFACE. TABLE XLIII. gives for any normal surface of frequency the per cent. of cases included between the average, 0, and any degree of deviation, the latter being measured in terms of the standard deviation of the distribution, dis. Tables XLIV. and XLV. give the same information when the degrees of deviation are in terms of the A. D.dis, and P. E.dis.. Thus the first line of entries of Table XLIII. reads: Between the average and .01 o either above or below, either + or -, there are .004 of the cases; between the average and +.02 there are .008 of the cases; between the average and — .03☛ there are .0120 of the cases, etc. = = It thus enables one to calculate the entire distribution of any trait which is normally distributed, the average and variability of which are known. For instance, if one finds for discrimination of color that the average 24.0 and the standard deviation 4.0, one finds. from the table that the ability 24-24.99 or that between the average and .250, will be possessed by 9.87 per cent. of the group; the ability 24-25.99 or that between 0 and +.50 by 19.15 per cent., and consequently the ability 2525.99 by 19.159.87, or 9.28 per cent. By thus finding the percentages included between the average ability and different amounts of deviation from it, and so between any two given limits of deviation from it, one gets, as the table of frequencies in our illustrative case, Table XLVI. This use of the tables gives a convenient means of measuring the degree to which the measures under investigation approximate to the probability curve distribution. If the table of actual frequencies of the measures is compared entry for entry with the frequencies given for corresponding deviations in the table for the probability curve, one can see at a glance the general closeness of correspondence. In making such comparisons the actual frequencies may properly be grouped so as to represent only 18 or more grades, and any most likely central point may be chosen with which to make the central point of the probability surface coincide. For example, let the measures in the first column of frequencies of Table XLVII. be the actual distribution. Their average is 76 —; their median, 76+; and their most likely mode, 75 or 77. 76 may be taken as the central point for the comparison. Their A. D. ais. from it is 2.65 steps (5.30 units). In the second column the actual frequencies are given in per cents. TABLE XLIII. From Table XLIV. TABLE OF VALUES OF THE NORMAL PROBABILITY INTEGRAL CORRESPONDING TO VALUES OF x/σ OR THE FRACTION OF THE AREA OF THE CURVE BETWEEN THE LIMITS 0 AND + x/σ OR 0 AND—x/6. 1.7 4554 4564 4573 4582 1.5 4332 4345 4358 4370 4383 4395 4406 4418 4429 4441 12 4545 10 4591 4633 9 4686 4693 4700 4706 7 4750 4756 4762 4767 6 4788 4794 2.0 4773 4778 4783 4799 4804 4808 4813 4817 5 4887 4890 2.5 4938 4940 4942 4943 4945 4946 4947 4949 4951 4952 2.6 4953 4955 4956 4958 4959 4960 4961 4962 4964 4965 2.7 4966 4967 4968 4969 4970 4970 4971 4972 4973 4974 1 2.8 4975 4975 4976 4977 4978 4978 4979 4980 4981 4981 2.9 4982 4982 4983 4983 4984 4984 4985 4985 4986 4986 14332 |