The ChiSquare 'Goodness of Fit' test is used to test whether a sample is drawn from a population that conforms to a specified distribution.
The hypothesis is:
H_{0} the sample conforms to the specified distribution
H_{1} the sample does not conform to the distribution
The test is illustrated by example. An organization has three categories of employees, 'A', 'B' and 'C'. It collects the following data:
Category

# Employees

Days Sick

A

100

10

B

60

12

C

40

14

Total

200

36

The organization wants to test the hypothesis:
H_{0} the proportion of sickness is the same for each category of employees
H_{1} the proportion of sickness differs between categories
The first step is to form the table. The 'expected' column shows the results that would be expected if the proportions were equal between categories ie. if the null hypothesis were true:
Category

# Employees

Days Well 
Expected

ChiSquare Contribution

Days Sick

Expected

ChiSquare Contribution

A

100

90

82.0

0.78

10

18.0

3.56

B

60

48

49.2

0.03

12

10.8

0.13

C

40

26

32.8

1.41

14

7.2

6.42

Total

200

164

164

2.22

36

36

10.11

The chisquare statistic is calculated by summing the chisquare contributions from each category:
Where:
A_{i} actual value for category 'i'
E_{i} expected value for category 'i'
There are two degrees of freedom (if two of the 'days sick' data values are known the third can be calculated from the totals).
The critical pvalue can be obtained from tables, or the pvalue can be calculated using eg. Excel:
=CHIDIST(12.33,2) gives 0.0021
Refer also to Contingency Tables for another application of the chisquare test.
