SEA-FISHERIES LABORATORY. 179 



Or the mean may be calculated, and then the 

 standard deviation, from the relations, mean = arbitrary- 

 origin plus or minus the first moment of the distribution 

 according to its sign; and standard deviation = square 

 root of the second moment of the distribution.* From the 

 standard deviation the " probable error " of the distribu- 

 tion may now be calculated from the relation, probable 

 error = 6745 \//* 2 . The values of the probable error are 

 now two values, one on either side of the mean, such that 

 there are as many observations within ordinates drawn 

 from the ^-values, as there are observations outside these 

 ordinates. If we are dealing with a distribution which 

 includes few observations, or which is a very irregular 

 one, this latter method is the only convenient one of 

 obtaining measures of dispersion or error, and the mean 

 is the only position-value which can be employed. 



The standard deviation may be regarded from two 

 points of view — either it is simply the square root of the 

 second moment of any frequency distribution; or it is a 

 parameter, and can strictly be utilised only with respect 

 to the normal curve of error. From this latter point of 

 view it is the pair of values of x which give us the points 

 of inflexion on the curve. It is the pair of values at which 

 the second differential coefficient becomes zero, and in the 



neighbourhood of which -j^ changes sign, and for the 

 normal curve of error these two values are given by 



But obviously we cannot regard any distribution as 

 that represented by the normal curve of error unless we 

 know that the causes which lead to the deviations expressed 

 by the distribution are also those which are considered in 



* The notation and methods adopted are those of W. Palin Elderton 

 " Frequency curves and correlation." 



