106 Francis B. Sumner 
measurements) 1s obtained by dividing the sum of all the terms 
of the series by the number of these terms. 
The standard deviation, which is, at present, most frequently 
employed as the measure of the variability of a series, is obtained 
by squaring all the individual “deviations” or departures from the 
mean, finding the sum, and then the average, of these squared 
deviations, and extracting the square root of the resulting average, 
1. €., 
4 sum of squared deviations 
number 
The reliability of the average or mean has, in each case, been 
indicated by the probable error, which is the number preceded by 
the + sign and annexed to the mean in the following tables. 
The value of this number is such that there is an even chance that 
the true mean (i. e., such as would be obtained from averaging 
an infinite number of terms) lies within the limits indicated by: 
given mean + probable error. The probable error of an average 
or mean 1s obtained by the formula 
-6745 > standard deviation 
V number of terms in the series 
The reliability of an average thus varies inversely as the varia- 
bility of the series and directly as the square root of the number 
of terms. ‘The probable error of the mean, here employed as an 
index of reliability, is not to be confused with the probable error 
of @ series, sometimes employed as an index of its variability. 
This latter is a number of such magnitude that it is exceeded by 
exactly one-half of the deviations. It has the value: .6745 x 
standard deviation. 
The reliability of the standard deviation, or figure denoting the 
variability of each series, is indicated by the probable error of the 
standard deviation. ‘This is obtained by the formula: 
-6745 > standard deviation 
V2 > number of terms 
