TYPE AND VARIABILITY 429 



Dividing 538.4103 by 327 after the manner of finding the 

 average deviation, we have the quotient 1.6465 ; but as the 

 deviations have all been squared during the operation it is 

 necessary to extract the square root of this number in order to 

 arrive at the correct value. The square root of 1.6465 is 1.28 +, 

 and this is the so-called standard deviation of the mathema- 

 tician, the universal sign for which is the Greek letter, small 

 sigma (a). 



Hence, to find the standard deviation, we have the rule : Find 

 the deviation of each frequency from the mean ; square each 

 deviation, and multiply by its corresponding frequency ; add the 

 products, divide by the total number of variates, and extract 

 the square root. 1 



Shortening the method. The large decimals can be avoided, 

 and the process of finding both the mean and the standard 

 deviation can be very much shortened, by assuming as a mean 

 the nearest probable measurement as determined by inspection 

 of the frequency distribution, and afterward making a suitable 

 correction. For example, in the present instance, we should 

 judge by inspection that the mean cannot be far from 9.0.* 

 This we infer from the fact that the distribution reduces both 

 ways from this point and quite evenly. Proceeding with this 

 assumption, denoting our "guess" by G and reckoning devia- 

 tion provisionally from this point, we have the result as seen 

 in the table on the following page. 



Considering first the mean : In column/(F G) we find x that 

 after multiplying the deviations from our assumed mean (9.0) 

 by their respective frequencies, the sum of the negative products 

 ( 181.0) exceeds the sum of the positive products (125.0) by 

 56.0 ; that is, the algebraic sum of the products is 56.0. 

 Our assumed mean is therefore too high by the amount of 

 - 56.0 -1-327!= o. 171. We then reduce our assumed mean 



1 Expressed in symbols the formula is <r \ _ 



H 



* The advantage of assuming this value from which to reckon deviation lies in 

 the fact that it is exact and contains but one decimal, while the true mean has at 

 least two decimal places, making relatively large numbers. 



t We divide by the total number (327) because we are dealing with a column 

 of products arising from the introduction of the frequencies. 



