(capital X is the Greek letter chi)
For example, does 787 tall to 277 short indicate a 3:1 ratio?
tall short
observed # 787 277
expected ratio 3/4 1/4
expected # (787 + 277)3/4 = 798 (787 + 277)1/4 = 266
(O - E) -11 +11
(O - E)^{2} 121 121
(O - E)^{2}/E 0.15 0.45
X^{2} = 0.15 + 0.45 = 0.60
Suppose we assume that we are sampling from a population with a 1:1
ratio
tall short
observed # 787 277
expected ratio 1/2 1/2
expected # (787 + 277)1/2 = 532 (787 + 277)1/2 = 532
(O - E) 255 -255
(O - E)^{2} 65,025 65,025
(O - E)^{2}/E 122.23 122.23
X^{2} = 122.23 + 122.23 = 244.45
Before determining the significance of the X^{2} value we must
determine the degrees of freedom. The degrees of freedom (d.f.) tell
us how many unique categories we have. In this case, we had 1064
plants, with 787 in the tall category, thus we have to have 277 in the
short category. The short category is not unique in that it can be
calculated from knowing the tall category.