44 WICKSELL, THE CORRELATION FUNCTION OF TYPE A, AND THE REGRESSION OF 1TS CHARACTERISTICS. 



Within the range of the observation the corresponding curve will be a parabola of 

 the second order. 



Pearson found (using moments to the fourth order) 



v y = 6,794 1 — 0,i 2 59(2/ — m 2 ) — 0,0" 7 e{y — ra.) 2 . 



His cubical formula involving moments to the sixth order was found to give no 

 better fit. 



A comparison of results and observation gives the following table. 



Tahle VI. 



Whorl 



x (Pearson) 



*„((S6) II) 



Observation 



First 



0,777 



6,867 



6,780 



Second 



6,854 



0,803 



6,813 



Third 



6,775 



6,680 



6,813 



Fourth 



6,541 



6,500 



6,48G 



Fifth 



6,151 



6,254 



6,172 



The curves will be seen in Plate 2. 



Example 6. Pearson's illustration B. Correlation between age and height of 

 head in 2272 girls. 



For the moments and correlation table we ref er to Pearson's memoir. The 

 characteristics needed for the regression of the head-heights on the age were 



ra, = 12,7007 years 



O x = 3,064 8 » 



(i 30 = — 0,006090 

 /?2i = + 0,030159 



T = +0,2941 



and we find 



ra 2 = 124,0467 mm. 

 a 2 = 6,9083 mm. 



/? 40 = — 0,012068 

 /? 3 , = — 0,022285 



r i0 = — 0,008087. 



r 30 = + 0,035533 



By (35) I we find, measuring y x in mm:s 



?/.,. — m 2 = 0,245470 +0,717663 (x — ra.) — 0,0261 33 (it — ra,) 2 — 0,001941 (x — Wl,) 3 . 



The corresponding curve Pearson found to be (including moments to the sixth order) 



y x — ra 2 = 0.280 J 94 + 0,7 2 28 66 (X — m,) — 0,02 9 58 (x — ra,) 2 — 0,002223 {x — »l,) 3 . 



The correspondence between our result, Pearson's result and the observation 

 is seen from the following table and from Plate 2. 



