c^{2}test is frequently used in biology
to find out whether observed data is different from expected results.
For example in a cross of flies, we found that 23 progenies are female
and 47 are male.
 Observed  Expected  Total 
Male  23  35  58 
Female  47  35  82 
Total  70  70  140 
For this simple example, there are only 2 columns (observed and expected) and 2 rows (male and female), thus
the degree of freedom
df = (number of columns  1) * (number of rows  1) = 1.
c^{2} is calculated using the following formula:
c^{2} = S
(Observed Value  Expected Value)^{2} / (Expected Value).
Note in this version of c^{2}test the expected value is calculated
from a hypothesis (i.e., the frequency of male or female progenies should be 0.5).
For the above example, c^{2} = (2335)^{2}/35 + (4535)^{2}/35
= 7.11.
Checking the c^{2} table

Probability 
Degrees of Freedom 
0.9 
0.5 
0.1 
0.05 
0.01 
1 
0.02 
0.46 
2.71 
3.84 
6.64 
2 
0.21 
1.39 
4.61 
5.99 
9.21 
3 
0.58 
2.37 
6.25 
7.82 
11.35 
4 
1.06 
3.36 
7.78 
9.49 
13.28 
5 
1.61 
4.35 
9.24 
11.07 
15.09 
we find p < 0.01.
Web
c^{2} calculator
