X² Short Formula

Suppose the ratio is X₁:X₂, then the proportions are:

X1	and	X2	;
X1+X2		X1+X2

    

Also, the expected numbers for each class would be

(n)	X1	and (n)	X2
	X1+X2		X1+X2

Back to our example, we have the ambiguous segregation ratio 1.9640:1 which can be converted to proportions:

1.9640	and	(1)
2.9640		2.9640

 

Then the expected number of each class are:

(n)1.9640	and	n(1)
2.9640		2.9640

The short formula for a X² of a 3:1 ratio is:

X2 =	[a1 - a2(3)]2	= 3.841
	3n

     
Now we know a₁ is the expected number of the dominant class and a₂ is the expected number of the recessive class. Then substituting the ambiguous expected numbers for each class gives:

X2 = 3.84 =	[	1.9640(n)	-	1(n)(3)	]	2
		2.9640		2.9640
	3n

Now we solve for n to find the critical F₂ family size such that more than r recessives indicates a 9:7 ratio and less than r recessives indicates a 3:1 ratio.

3.84 =        n²        x   (1.96-3)² x   1  
            (2.9640)            1         3n

3.84 =   (1.04)²   x  n 
          (2.9640)²     3

n = 94.31

We must grow 95 plants to distinguish a 3:1 ratio from a 9:7 ratio.

Let n=94, we use the ambigous ratio to derive the observed class and test with a X² to try to distinguish the two ratios.

The dominant class will have the observed number

(95)  1.964  = 62.9~63.
        2.964

The observed number for the recessive class
(95)      1    = 32.
        2.964