Mathematical Study of Suppia' axd Demand 85 



April, 100 ; May, 99 ; June, 97 ; July, 98 ; August, 90 ; September, 

 93; October, 108; November, 103, and December, 106. 



After securing normal seasonal variation, the next step is to 

 modify secular trend for seasonal variation. Secular trend of hog 

 prices, as modified seasonally, is portrayed in Chart II. The sec- 

 ular trend price of hogs in January, 1903, is $.5.19, which sum, 

 nmltiplied by the seasonal factor 96, gives $4-. 98 as the secular 

 price of hogs modified seasonally for January, 1903. The actual 

 price was $6.60, or $1.62 above the secular modified seasonally, or 

 31 per cent greater than the secular price of $5.19. In this way 

 the percentage of departure for each month from 1903 thru 1916 

 may be figured. This has been done for hog prices, hog receipts 

 and bank clearings outside of New York City.* 



Now, as it happens, hog receipts are a much more violently 

 fluctuating series than bank clearings outside of New York City. 

 To put the series on an even footing, resort is made to what is 

 known as the standard deviation. To secure the standard devia- 

 tion of hog price percentage departures, add up the squares of 

 these departures. The total for the 168 months from 1903 thru 

 1916 is 31,894, or, dividing by 168, we get 190. The square root 

 of 190 is 13.8, which is the standard deviation of hog prices. 

 Standard deviation means that the probabilities are that on the 

 average not more than one out of three of the series of figures an- 

 dcr consideration will exceed the standard deviation. Standard 

 deviation for hog receipts is 15, and for bank clearings 8.7. This 

 indicates that hog receipts depart from the secular trend as modi- 

 fied seasonally with nearly twice as great violence as do bank 

 clearings. 



To put all three series on the same footing, we divide the per- 

 centage departures by the standard deviation, 13.8 in the case of 

 hog prices, 15 in the case of hog receipts, and 8.7 in the case of 

 bank clearings. In January of 1903, for example, hog prices 

 were greater than the secular modified seasonally by 2.3 times the 

 standard deviation ; hog receipts were less b}^ .3 of the standard 



*Warren M. Persons, in a footnote on page 35 of the January, 1919, 

 Review of Economic Statistics, expresses the method of ascertaining per- 

 centage departure from the secular trend in mathematical symbols as 

 follows: "l^t the oriirinal series heyriniiin^r witli January be Xi, X2, Xs, 

 . . . . X„, the ordinates of secular trend be d, O2, O3, .... On, 

 anil the adjusted indices of seasonal variation for twelve months be 

 '^it f?2. ^3) • . ■ ■ !^i2 per cent, respectively. Then the items for secular 

 trend and seasonal variation are: 



X, -S, O, X2-P, O2 X3-S3O3 X3-S. 0„ 



O, ' Oa ' O, ' ' " ' 0,3 etc." 



