WEATHER AND CHANGE IN WEIGHT OF BEE COLONY 7 



It may not be amiss at this place to give, for those not well versed 

 in biometrics, a brief explanation of the terms "correlation" and 

 -" probable error." The following quotation is taken from Bowley 

 (8, p. 816) : 



When two quantities are so related that the fluctuations in one are in sympa- 

 thy with fluctuations in the other, so that an increase or decrease of one is found 

 in connection with an increase or decrease (or inversely) of the other, and the 

 greater the magnitude of the changes in the one, the greater the magnitude of 

 changes in the other, the quantities are said to be correlated. 



We may have either a positive or a negative correlation. When 

 a change in one quantity or variable is accompanied by a direct change 

 in the other the correlation is said to be positive. A perfect positive 

 correlation has the value of 1. When the correlation is less than per- 

 fect it must be written as a decimal of one, such as .75. When the 

 relationship between two quantities or variables is indirect, such as 

 an increase in one being accompanied by a decrease in the other, the 

 correlation is negative. A perfect negative correlation has the value 

 of — 1. Such a relationship less than perfect must also be written as 

 a decimal and is always preceded by a minus sign. Coefficients of 

 correlation state in numerical terms the relationship between two 

 variables. Graphs are useful in showing relationship between two 

 variables, but they do not give numerical correlation values, and it 

 is often difficult from the study of a graph to discover slight relation- 

 ship or entire absence of relationship; this difficulty does not exist 

 in the case of correlation. It is often convenient to think of a coeffi- 

 cient of correlation in terms of percentage ; thus, a correlation written 

 0.8654 may be read as 86.54 per cent. 



The probable error is a term applied in biometrics to make correc- 

 tions in a calculation where complete data are lacking. Sample 

 measurements must be made preliminary to a biometric calculation 

 and it is rarely possible to obtain a complete series of samples. For 

 instance, in calculating the correlation between temperature and the 

 change in colony weight we may have the changes in weight occurr- 

 ing at 80°, 81°, 82°, and 84° F., the change at 83° F., being for some 

 reason impossible to secure. The probable error gives the measure 

 of unreliability due to lack of sufficient data k Obviously, the smaller 

 the number of data involved the greater the probable error. The 

 probable error is written with a combined plus ( + ) and minus ( — ) 

 sign (±), and represents the true correlation as falling somewhere, 

 either above or below the calculated correlation, by a difference 

 most probably equal to the value of the probable error. A cor- 

 relation written .7500 ±.0600 indicates that there is an even chance 

 that the true value lies between .7500 4- .0600, or .8100, and .7500- 

 .0600, or .6900. To be significant, the coefficient of correlation should 

 be at least about four times its probable error. When it is less than 

 this the correlation approaches zero in its significance and is of impor- 

 tance primarily as showing whether a relationship is positive or nega- 

 tive. In discussing the probable error Yule (86, p. 811) says: 



If an error or deviation in, say, a certain proportion p only just exceed the 

 probable error, it is as likely as not to occur in simple sampling; if it exceed 

 twice the probable error (in either direction), it is likely to occur as a deviation 

 of simple sampling about 18 times in 100 trials — or the odds are about 4.6 to 1 

 against its occurring at any one trial. For a range of three times the probable 

 error the odds are about 22 to 1, and for a range of four times the probable error 



