6 MAX LEVINE 



small or great. The closer the individual organisms group themselves 

 about the mean, or average, the smaller the standard deviation. 



An example may make clear the meaning and significance of the standard 

 deviation. Suppose that the amounts of acid formed by a group (A) of 4 

 organisms in glucose broth are 2.1, 2.2, 2.2, and 2.3% normal acid, and that 

 those formed by another group (B) of 4 organisms are 1.9, 2, 2.4, and 2.5% 

 normal acid. The average for each group is 2.2, but mere inspection shows 

 that the organisms in Group A and those in Group B are quite differently 

 distributed with respect to this average. In large collections of data inspection 

 is impracticable, but the standard deviation serves well in its place. The 

 standard deviation in Group A is 0.07 while for Group B it is 0.25. The 

 larger deviation in B denotes that the individuals in -the group wander farther 

 away from the average than do those in Group A. 



The probable error is employed to indicate what confidence is to 

 be placed in statistical results. The reliability of the mean and stand- 

 ard deviation may be determined by calculating their probable errors, 

 but in this paper only the probable error of a single variate is con- 

 sidered. In a normal distribution the probable error of a single 

 variate of a series of observations is denned as that departure from 

 the mean, on either side, within which exactly one-half of the variates 

 are found ; that is, if in the study of acid-production by a large number 

 of organisms, it is found that the mean (average) amount of acid 

 formed is 2.25% normal, and that the probable error of a single obser- 

 vation is 0.15, then half of the organisms have formed between 

 2.1% and 2.4% normal acid. 



The coefficient of variability is the ratio of the standard deviation 

 to the mean (^ ). It is an abstract measure of variability and may 

 therefore be employed to advantage for comparing variability among 

 different characters, or in the same character among different groups 

 of organisms, particularly if their means differ widely. 5 



ACID-PRODUCTION IN SUBSTANCES FERMENTED BY ALL OF THE 

 TEST ORGANISMS 



Glucose, galactose, mannitol, maltose, and lactose were decomposed 

 with gas-production by all strains. 



A. GLUCOSE 



The frequency distributions of the organisms with respect to acid- 

 formation from glucose are shown in Table 1, where the relation of 



8 For a detailed description of these constants the reader is referred to Principles of 

 Breeding (1907), by E. Davenport; Statistical Methods (1904), by C. B. Davenport; Precision 

 of Measurements (1909), by Goodwin, and to an Introduction to the Theory of Statistics 

 (1916), by Yule. 



