408 W. A. BECKER AND M. A. MARS DEN 



Therefore 



2 1 1 2 



o^ = — pa- — Oj 

 p m r ^ m a 



where — p q is the binomial sampling variance. 



The variance of the proportion therefore is less than the standard 



binomial variance by lo 2 ,. 

 7 m d 



ARC SINE SQUARE ROOT TRANSFORMATION 



If p is the average probability of a seedling within a plot being 

 healthy then the number of healthy seedlings is n = m p, the expected 

 value of n, E(n) = u. 



The binomial sampling variance m p (1 - p) is a function only of the 

 mean probability of a seedling being healthy, y. The standard deviation 

 (0^) is a function of y, 



a y = <Kv) = [p(1 - ™ -1 y)] 1/2 • 



We can remove this dependence of the variance on the mean if we can find 

 a function of the mean /(p) = Z, such that the new variable Z has a con- 

 stant variance (Scheffe', 1959, p. 365). Let the standard deviation of Z be 

 equal to the standard deviation of y, times the derivative of the func- 

 tion /(y), 



o z =O y /'(y), 



Then 



/• (y) = c z /a^ = o z / <Ku) 



The function /"(y) which gives a new variable Z, constant variance can be 

 obtained by the integration of f } (y) , 



f(v) = Irtv) du = o ? I ((.(y)" 1 du. 



Substituting the estimate p for m y will give 



f (p) = o z / m p (1 - p) d p , 



o 1/2 ■ r^l/2 



= 2 #7 o arcsm (p) + e 



The above function can be simplified by taking a - 0, and choosing a ? = 

 28.65 m 1 / 2 . 



f (p) - arcs in (p) 



for the arcsin in degrees 



