= Av. 10 in one direction or the amount of divergence in one direction, more divergence than which has a given degree of improbability. The same methods serve if the unreliability is of a variability or of a difference or of a relationship-in short, for all cases where the unreliability is measured by the variability of a divergence of true from obtained, and this divergence is distributed in a normal probability surface. The following problems will offer opportunity for acquiring selfconfidence in the use of the tables in connection with all sorts of questions about unreliability : = 56. t.-o. Av. 1.6. (a) What is the probability of a difference between Av. and Av... of 4.0 or more? (b) What are the chances that Av... will be 3.2 greater than Av...? (c) Between what limits will the true average lie with a probability of 9999 to 1? 57. ☛t.-o. var. = .4. (a) What is the probability that the true variability is more than .8 less than the obtained? (b) That the true variability is not more than .6 above or below the obtained? = 58. t-o. diff..5. The actually obtained difference is, Av., Av., 1.2. (a) What is the probability that the true difference is zero or less than zero? (b) That the true difference is: Av., - Av.2 = 2.4 or more? (c) That the true superiority of Av., over Av., is between 1.7 and .7? (d) What limits would you assign for the true difference to be sure that the chances would be 20 to 1 against their being exceeded? 59. ro. = +.48. ☛t.-o. rel. = .04. (a) Between what limits does the true relationship lie with practical certainty (it is customary to take 997 out of 1,000 as practical certainty)? (b) What is the chance that the true relationship is as low as .40? 60. Av.。. = 22.6. A.D.t.-o. Av. = .4. (a) What is the chance that the true average is as large as 24.0? (b) That it is as small as 22.0? 61. Av... =28.2. P. E.t.-o. Av..6. that the true average is less than 26.0? Av... by less than 2.0? · (a) What is the chance (b) That it varies from 62. If it were true that the chances were 82 to 18 that the true average would not vary from the obtained by more than 13.4, what would be the value of P. E.t.-o. Av. ? = 63. Av.1 = 10.1, Av.2 12.4. P. E.t.— o. diff. of Av.1 and Av.2 = = 1.0. (a) What are the chances that Av., - Av., 0 or less? (b) 1.0 or less? (c) 2.5 or more? (d) Between 2.0 and 2.8? (e) Between 1.0 and 3.3? 64. P. E.dis. obt. = 1.6, A. D.t.-o. var. = 0.1. (a) What are the chances that P. E.dis. will be between 1.4 and 1.8? (b) That it will not exceed 1.9? (c) What limits must be taken such that the true P. E.dis. will be practically certain (see question 59) not to exceed them? = .008. 65. r. + .39, P. E.t.-o. rel. true relationship being as high as + 40? As +.50? What is the chance of the As + 41? As + .42? 66. Speaking roughly, the true measure is practically certain to lie between the following limits: Justify this statement from the tables. 10. r2 = .04, P. E.t. 67. Ti = t.-o. diff. r1 and r2 .06. (a) What is the chance that the true r2 is really equal to or greater than the true r1? (b) What is the chance that the true r, is greater than the true r2? 2 X Given the fact that two groups are normally distributed and that the central tendency of the first is X plus the central tendency of the second, X being in terms of the variability of the first, what per cent. of the first group will exceed the central point for the second? The per cent. will equal 50 plus the per cent. included between the central point and a point X above it. (See Fig. 86.) This is, of course, given directly by the table. For instance, let group 1 have 4. and dis. Let group 2 have Av. = 8. The difference + 2 equals .50 (of distribution of group 1). The percentage of group 1 exceeding the average for group 2 will be 50+ 19.15 or 69.15 per cent. When the first group is inferior to the second, the calculation is the same, replacing 50 per cent. plus by 50 per cent. minus. = 68. If boys in spelling average 18.6 with odis. 2.4, and girls average 20.0, what per cent. of boys will reach or exceed the average for girls? 69. If the per cent. of attendance to enrollment in cities averages 74 with a P. E.dis, of 8.6, and the same trait in towns averages 64, what per cent. of cities will reach or exceed the average for towns? 70. If the median strength of 10-year-old boys is 16.2 with adis. = 2.1, and the median strength of 11-year-old boys is 17.4, what per cent. of 10-year-olds will be stronger than the median 11-year-olds? CHAPTER XII. SOURCES OF ERROR IN MEASUREMENTS. So far our supposition has been that the measures with which we start are accurate representatives of the fact measured, that A really did misspell the word which we score misspelled, that B did really take the .150 sec. to react which the chronoscope recorded, that the school enrollment and average attendance given for cities in the U. S. Commissioner's report give the real facts, that the number of children recorded in certain genealogy books for certain families were the real numbers. Our problem has been to make the best use of the data and introduce no error in manipulating them. But that a measure should thus perfectly represent a fact, the fact must be measured by a perfect instrument used by an infallible observer. In reality, any measure is a compound of a fact and the errors which the instrument and observer will surely make. A constant error is A watch that is make the attend These errors may be constant or variable. one tending more in one direction than the other. too slow, a tendency of school superintendents to ance record too high, are examples. Variable or chance errors are those tending in the long run to make the amount lower as often and as much as higher. The unevenness in action of a delicate balance due to dust, air currents, etc., the errors in addition made by the clerks in a superintendent's office, are examples. Variable errors do not make any measure unfair, but only less exact and less reliable. If a body is weighed by an instrument which fluctuates so as to give 156.1, 156.2, 156.3, 156.3, 156.3, 156.3, 156.4, 156.4 and 156.4 in nine measurements, but is known not to weigh too light or heavy, 156.3 is a true measure, but the 156.3 only means between 156.25 and 156.35 and there is a slight chance of its being 156.2 or 156.4 (about 1 chance in 500). If, on the contrary, a body is weighed by an instrument which fluctuates so little as to give 156.298, 156.299, 156.300, 156.300, 156.300, 156.301, 156.301 and 156.301, and which is known not to weigh too light or heavy, the 156.300 means between 156.2995 and 156.3005 and there is now certainty that the measure is not so low as 156.2 or so high as 156.4. Indeed, there is certainty that it is between 156.298 and 156.302. There is no great advantage in decreasing the amount of the variable error by using more delicate instruments or more care in observing, unless the precision and reliability thereby obtained can be preserved in the further use of the measurements. The advantage that there is consists in the moral and intellectual training one gets and in the possibility that the measures may later be used for purposes other than one expects. If we wish to get A's average error in trying to equal a 100-mm. line, measurements may be made with the aid of a glass tomm., but the variation between A's separate trials is so great that the larger error due to measuring each line so roughly as into mms. is insignificant. Indeed, measurements to a millimeter really do as well. If we wish to compare the reaction time of 1,000 boys with that of 1,000 girls, the median of 10 times being taken for each individual, measures in hundredths of seconds will do as well as measurements in thousandths. Much time may be wasted in refining measurements in cases where no advantage accrues. And much ignorance is shown by the many students who disparage all measurements that are subject to a large variable error. They either do not know or forget that the reliability of a measure is due to the number of cases as well as to their variability, and that in the more complex and subtle mental traits it is always practicable to increase the number of measurements, but often impossible to make them less subject to variable errors. They also forget that the natural and real variability of the fact itself is often so large as to make the variability due to errors of instruments and observation practically negligible. Constant errors, on the other hand, are never negligible. The errors we make in interpreting handwriting would not, in a comparison of 1,000 boys with 1,000 girls in spelling ability, be worth spending a day on, even if thereby one could rectify them all, but if the teachers of the girls pronounced the words more clearly and phonetically than those of the boys, it would be necessary to discuss the proper discount or give up all hopes of precision. That a genealogist by mistake sometimes writes 4 or 7 matters practically nil to the student of vital statistics, but the genealogist's constant |