June 12, 1914] 



SCIENCE 



879 



This difference problem marks a new and very 

 probably most important departure in statistical 

 theory. 



Clearly a knowledge of the average differ- 

 ence in scholarship of adjacent individuals 

 supplied by Pearson's formulas involves also 

 a knowledge of the average difference in 

 scholarship between any two individuals. We 

 shall display in our tables the difference be- 



difference between the tenth and eleventh. 

 Similar statements apply to the poorest pupil 

 and the one next above him. These relations 

 are brought out by the adjoining figure. 



Distances measured to the right and left 

 of the zero point signify abilities above and 

 below the modal ability. The relative stand- 

 ings of the members of an average class of 

 20 are indicated by the dots. Observe the 



Pig. 1. 



tween the modal or most frequent scholarship 

 of the class and the scholarship of any indi- 

 vidual in the class. 



The columns headed " mark/s " signify the 

 ability of the pupil above or below the modal 

 ability, divided by s, the standard deviation 

 of the total group of students (say first year 

 high school students) from which the par- 

 ticular class is taken at random as a " fair 

 sample." It will be noticed that a large stand- 

 ard deviation indicates a large range of dis- 

 tribution — that is, a large difference of accom- 

 plishment between the best and poorest in the 

 class. In freshman classes the standard 

 deviation is apt to be large, because of great 

 difference in preparation. For our purposes, 

 the exact value of the standard deviation i3 

 of no interest. We are concerned more with 

 the ratios of differential abilities than with 

 their absolute values. Hence we shall take 

 s ^ 1, or, if more convenient, s ^^ 10. 



Consider a class of 20 pupils. The modal 

 or " mediocre " ability is taken here, as in the 

 other cases, as the standard of reference and 

 is marked 0. Abilities of students are ar- 

 ranged symmetrically above and below, and 

 marked positive and negative. By subtract- 

 ing the ability of a pupil of rank n from that 

 of his neighbor below, we get the differential 

 ability of the two. In a class of twenty the 

 difference in average ability between the tenth 

 and eleventh pupil is .13. The difference be- 

 tween the first and second pupil is .5. Thus 

 the difference between the first and second 

 pupils is about four times greater than the 



denseness of the dots near the modal position 

 and the isolation of those at the ends. 



When the number of pupils in a class ia 

 larger, the differential ability of the pupils 

 ranking next to each other becomes smaller. 

 Thus in a class of 100, the difference between 

 the first and second is on an average .3, that 

 between the 50th and 51st is on an average 

 .02, but the former difference is about 15 times 

 greater than the second. The importance of 

 these relations is brought out by Pearson in 

 the following words : 



It is not possible to pass over the general bear- 

 ing of such results on human relations. If we 

 define "individuality" as diflference in character 

 between a man and his compeers, we see how im- 

 mensely individuality is emphasized as we pass 

 from the average or modal individuals to the ex- 

 ceptional man. Differences in ability, in power to 

 create, to discover, to rule men, do not go by uni- 

 form stages. We know this by experience, but we 

 see it here as a direct consequence of statistical 

 theory, flowing from a characteristic and familiar 

 chance distribution. We ought not to be surprised, 

 as we frequently are, at the results of competitive 

 examination, where the difference in marks be- 

 tween the first men is so much greater than occurs 

 between men towards the middle of the list. In 

 the same way the individuality of imbeciles and 

 criminals at the other end of the intellectual and 

 moral scales receives its due statistical apprecia- 

 tion. 



The total range of distribution for classes 

 of random pupils not exceeding 100 is about 

 2.5s on each side of the modal line, where s 

 is the standard deviation. Taking s = 1 or 

 s = 10 we have a scale for marking, the objec- 



