190 Appendix 



correct the guess by a simple process. (Moore, — Forecasting 

 the Yield and Price of Cotton p. 19. Theorem II.) 



2. The averages of two series, however, tell us very little. 

 We need still another measure, in terms of which, we can tell, 

 in each single case the proportion in which the variation of 

 one observation from the average of its series stands to the 

 variation of another observation from the average of its series. 

 The best description of the need and derivation of such a meas- 

 ure is found in Moore's "Forecasting the Yield and Price of 

 Cotton," pages 20-22. 



"The arithmetical mean of the frequency distribution gives us 

 one of the most important summary descriptions of the dis- 

 tribution: it gives the centre of density of the distribution. 

 But in economic, as well as in most other measurements, it is 

 extremely important to know how the several observations are 

 grouped about the arithmetical mean of the measurements, and 

 a coefficient showing the manner of grouping is a measure of 

 dispersion. Just as we found that the arithmetical mean of the 

 measurements gives us an idea of the centre of the density 

 of the measurements, so, as a measure of dispersion, we might 

 take the arithmetical mean of the deviations of the magnitudes 

 from the mean of the observations. But if we followed this 

 plan, we should meet with an embarrassing difficulty: The 

 deviations of the measurements from the arithmetical mean 

 are some of them positive and some of them negative, and if 

 we take account of the signs of the deviations, then, the sum 

 of the deviations is zero. We therefore choose, as our measure 

 of dispersion the square-root of the mean square of the devia- 

 tions about the arithmetical mean of the observations and we 

 call this measure the "standard deviation." The measure of the 

 dispersion of a series of observations about its average is then 

 derived by squaring the deviation of each observation, summing 

 the squares and dividing by the number of observations and 

 extracting the square root. With 2 as our symbol for "the sum 

 of," and n for the number of cases, in a series whose individual 



2 X 2 

 deviations are designed by Y, the standard deviation is: 



For the example the total of column 6, table 17 gives the 

 sum of the squares of the population deviationsl56,428,000.When 

 this sum is divided by n (100) and corrected for the difference 

 between the guessed and true averages the result is 11,49,035. 



