218 



F. E. LUTZ. 



why they did not apply this simple test. It is called the standard 

 deviation and must be known to everyone who has ever done 

 any statistical work. The phrases just quoted are taken from 

 the discussion of the counts of the hooks on the hind wings. 

 Let us therefore examine them by means of this confessedly bet- 

 ter measure. 



We find that for the drones we have : 



For the workers we have : 



Lot. 



No. of Specimens. 



I. 



II. 



III. 



50 



35 

 100 



Standard Deviation. 



1.5223 

 I-55 6 4 

 I-5523 



Probable Error 



o. 1027 



0.0397 

 0.0740 



This gives an average standard deviation, or variability, for 

 the drones of 1.8592 0.1028; and for the workers of 1.5437 

 0.0721. But we see that the difference between the averages 

 for the two sexes is less than the difference between the two sets 

 of drones from the same hive (I. and III.) ; and, considering the 

 probable errors, neither is significant. If we omit the three small 

 series because of their large probable errors, we see that the dif- 

 ference between the variabilities of the two sexes is even smaller 

 and clearly not significant. It is also unfortunate that the work 

 should have been passed by both the authors and still two of the 

 nine averages be wrong. The average for lot II. of the workers 

 is 20.99, n t 21.08 ; and that for lot VI. of the drones is either 

 22.65 or 22-75, according as we do or do not include the indi- 

 vidual with 12 hooks, but it is surely not 22.42. This was prob- 

 ably gotten by including this individual (although he was excluded 

 by their argument above), and then using 100 as the total num- 

 ber, but for some strange reason there are only 99 of the 100 

 said to have been studied which are listed. 



