ON MENTAL AND PHYSICAL FACTORS INVOLVED IN EDUCATION. 30t> 



the boys and girls of an ordinary mixed school in a ' moderate ' neigh- 

 bourhood; and the boys and girls of the junior mixed and elder (girls) 

 departments of a special school for the mentally defective. 



Numbers. (Table I.) 



In all, 936 ' normal ' children and 111 ' defective ' children have 

 been examined in London with the same series of tests (series 11). 

 Taken in isolation, the numbers in some of the age-groups are small. 

 Those above 13 and below 8 years of age are so few, and so highly 

 selected, as to be negligible for general comparison. 



Age-Averages. (Tables II. and III.) 



The results, for the most part, show a steady progress from year to 

 year. The average rate of progress in addition, subtraction, multipli- 

 cation, and division, is about 3, 5, 7, and 4J marks per annum respec- 

 tively. At 10 years, the children attain on an average 22, 44, 40, and 

 26 marks: i.e., they work at the rate of 8 or 9 operations a minute 

 in multiphcation and subtraction, and at about half that rate in 

 division and addition. At the age of 11 the rate of progress declines ; 

 and at 13 the average may even fall. In this decline, an important, 

 but not the sole factor, is doubtless the transference of the best scholars 

 to secondary and central schools. If we assume that, with a complete 

 sample for the higher ages, progress would continue at nearly the same 

 rate, then the following regression-equations would serve to calculate 

 very approximately the norm from the age last birthday : — 



Addition-mark = 4 x age — 18 



Subtraction-mark = 8 x age — 36 

 Multiplication-mark = 10 x age — 60 

 Division- mark = 8 x age — 54 



Standard Deviation (Table IV.) and Range (Table V.). 



Within each age, the variation of individuals is considerable. The 

 standard deviation increases absolutely with increase of age, but 

 diminishes relatively to the age-average from about half the average to 

 about a third. 



The best performance at the age of 9 usually surpasses the average 

 performance at the age of 13 ; the worst performance at the age of 13 

 usually falls to the average performance at the age of 8. The best 

 performance in any age-group is double the average for that group, but 

 occasionally may be four or five times as large. 



Overlap of Ages. (Figure 1.) 



Owing to the wide individual variation, the overlap between the 

 several ages in any single test is enormous. The knowledge of the 

 ' norm ' or average for a given age, without knowledge of the amount 



