studies in Arithmetic, 1916-1917 7 



ponding frequencies of the several distributions were added 

 together for a total distribution for that grade and the city 

 medians were calculated from this distribution. The city dis- 

 tributions for seventh grade division are given in Table I. 

 The median number of examples right was obtained by multi- 

 plying the median number of attempts (rate) by the median 

 accuracy. 



It will be remembered that Class Record Sheet No. 1, 

 which is with Series B, does not provide a distribution for the 

 per cent of accuracy when it falls below 50. In all cases 

 where the median accuracy of a class fell below 50 per cent 

 the original test sheets were distributed according to rights 

 only to obtain the median for rights. This latter median was 

 divided by the class median in attempts to get the per cent of 

 accuracy or dependability. In some cases it was not pos- 

 sible to compute the median rights and accuracy because a 

 few schools did not send in the individual test sheets. For 

 example, in Tables II to VI, this is the reason why no scores 

 are given for City 4 in the sixth grade addition, fifth grade 

 addition and division, and the fourth grade in all the funda- 

 mental operations. 



Calculation of State Scores. The calculation of state 

 scores was similar to that of the city scores. The frequencies 

 of the several city distributions were added to form a state 

 distribution. The median of this distribution is the state 

 score. The method is illustrated in Table I. 



Importance of This Method of Calculating City and State 

 Medians. Sometimes the median of two or more distribu- 

 tions is found by averaging the medians of the several dis- 

 tributions. The validity of this method is based upon the as- 

 sumption that each of the distributions contain the same num- 

 ber of pupils. When this is not the case, the ''average" me- 

 dian is not the true median. The difference in the results ob- 

 tained may be illustrated by the addition medians for num- 

 ber of examples attempted. Carried to two decimal places 

 they are 4.97, 6.33, 7.01, 7.67, and 8.60 (see Tables II to VI), 

 whereas the ''average" medians are 5.05, 6.27, 7.16, 8.03, and 

 8.65. In four cases the "average" medians are larger. 



The reason for these differences is that when city medians 

 are averaged, each city contributes equally to the "average' 

 median. When the distributions are combined, each city con- 



