The χ^2 value is calculated for each cell in a contingency table. It is calculated as the
difference between each observed and corresponding expected count squared (this
squared difference makes all values positive or zero) and then divided by the expected
count (this standardizes all values). Each cell’s contribution to χ^2 is then added to provide
an overall χ^2 statistic for the contingency table. The degrees of freedom are calculated as
(rows−1×columns−1) which is 1 in a 2×2 table.
The χ^2 statistic in notational form is:
Chi-
square
—6.1where O is the observed cell frequency and E is the expected cell frequency.
Data from the first example (Figure 6.1) on the effect of the MEd programme on
teachers’ understanding of educational issues is used to illustrate calculation of the
overall χ^2 statistic. For clarity of presentation each cell in the table is labelled A to D.
(The number of females has been changed to seventy-nine, as there was an error in the
original paper. See earlier comment, p. 169.)
(^) Understanding of Educational Issues
(^) An increase reported No increase reported
Gender: Total
Male 21 (A) 27 (B) 48
Female 10 (C) 69 (D) 79
Total 31 96 127
Computational steps:
1 Calculate expected values for each cell A to D
Cell A: Expected=(48×31)/127=11.717
Cell B: Expected=(48×96)/127=36.283
Cell C: Expected=(79×31)/127=19.283
Cell D: Expected=(79×96)/127=59.717
2 Calculate the value of χ^2 for each cell A to D. Use formulae 6.1
Cell A: χ^2 =(21−11.717)2/1 1.717=7.355
Cell B: χ^2 =(27−36.283)^2 /36.283=2.375
Cell C: χ^2 =(10−19.283)^2 /19.283=4.469
Cell D: χ^2 =(69−59.717)^2 /59.717=1.443
3 Sum all the χ^2 values
= 7.355+2.375+4.469+1.443
Total χ^2 =15.64, df=1Inferences involving binomial and nominal count data 169