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 (ignore the 'expected' and 'chisquare' contribution columns for the moment):
Category

# Employees

Days Sick

Expected

ChiSquare Contribution

A

100

10

18

3.56

B

60

12

10.8

0.13

C

40

14

7.2

6.42

Total

200

36

36

10.11

If the sample conformed exactly to the distribution, the days sick would be shared out as shown in the expected column. 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(10.11,2) gives 0.0064
Contingency tables are an application of the chisquare test used when the relationship is between two variables. For example, the organization decides to investigate whether there is a relationship between employers who take sick leave, and who take their full entitlement of annual leave. The hypothesis is:
H_{0} there is no relationship between taking leave and propensity for sickness
H_{1} there is a relationship between taking leave and sickness
The data are as follows:

Sick

Not Sick

Total

Take Leave 
65

55

120

Don't take leave 
50

30

80

Total 
115

85

200

The expected values for the individual cells are found from:
The chisquare contributions for each cell are calculate from:
The expected values and the chisquare contribution are

Sick

Not Sick

Total

Take Leave 
69 (0.23)

51 (0.31)

120

Don't take leave 
46 (0.35)

34 (0.47)

80

Total 
115

85

200

The total chisquare value is 1.36. The number of degrees of freedom can be calculated from:
(rows  1) x (column  1)
This gives one degree of freedom. The number of degrees of freedom may also be obtained by considering that given any cell and the totals, the values in the remaining cells can be calculated.
From Excel =CHIDIST(1.36,1) the pvalue is 0.24; this would not be accepted at the 0.05 level of significance.
