234 BIOLOGICAL EFFECTS OF RADIATION 



three-quarters of the observations occur. These Umits have, however, 

 the same difficulty of determination as the median. 



Since the mean and standard deviation are the most commonly used 

 constants in summarizing a series of measurements, the discussion in this 

 section will be limited to the treatment of these two constants. Before 

 proceeding to the determination of these constants for a set of observa- 

 tions, it is usually convenient to order the observations into what is 

 called a frequency distribution. A frequency distribution is nothing 

 more than a table giving values on the scale of measurement and opposite 

 each the number of individuals found to have that particular scale value. 

 If the unit of measurement is very small compared with the degree of 

 variation between the smallest and largest individuals under discussion, 

 such a frequency table becomes very long and it is customary to group 

 the observations into classes, giving in the frequency distribution the 

 numbers of individuals falling within continuous evenly spaced classes 

 on the scale. Grouping of this type amounts to nothing more than a 

 decision that the original measurements were taken with too fine a scale 

 division, and that they might just as well have been made on a scale 

 whose unit was equal to the unit of grouping. 



Some of the various treatments that can be given a frequency dis- 

 tribution and the determination of the mean and standard deviation 

 may perhaps best be presented by analyzing some actual data and we 

 shall take for this purpose measurements from a paper by Muriel Robert- 

 son (6) in which she compared the length of the protozoon Bodo caudatus 

 in a sample subjected to gamma-ray radiation with that of a nonirra- 

 diated group. Five hundred bodos were included in the irradiated and 

 the same number in the control group, and their lengths were measured in 

 units of 0.5ju. These measurements arranged in frequency distributions 

 are shown in Table 5, the first column giving the scale of the measure- 

 ments in 0.5ai and the second column the number of individuals having 

 the length indicated. The distributions are presented graphically in 

 Fig. 2. 



The mean and standard deviation of these 500 items might be com- 

 puted by following the definitions directly, but the amount of labor can 

 be very much reduced by using the frequency distribution and following 

 the process given below. An arbitrary scale labeled x is selected, the 

 origin of this scale being placed at any point, but for convenience near 

 the center of the distribution, and with this new scale the columns xf 

 and x-f are computed as indicated. The mean and standard deviation 

 can then be obtained by the following equations: 



y,xf 

 Mean = m = origin -|- -=qr 



