Chi-Square Distribution [Properties]

Chi- Square is an important non-parametric test and as such now rigid assumptions are necessary is respect of the type of population we require only the degree of freedom for using this test. Non- parametric chi-square test can be used 

  • As a test of goodness of fit
  • As a test of independence 

This test was developed by statistician “Karl Pearson”.

As a test of goodness of fit: Chi – square test enable us to explain how does the theoretical distribution fit to the observed data. If the calculated value of the chi-square is less then the table value at a certain level of significance which means that there is no difference between observed and expected frequencies otherwise there is significant difference between expected and observed frequencies.

As a test of independence : Chi-Square test enables us to explain weather or not two attributes are associated. For example we may be interested in knowing weather a new medicine is effective in controlling fever or not. Chi – square test will help in deciding this issue. For this first of all we calculate chi – square value and the value is compared with the table value. 

If the calculated value of chi – square is less than the table value at a given level of significance for given degree of freedom we conclude that the two attributes are independent i.e medicine is not effective in controlling fever.

We may apply the chi – square test either as a test of goodness of fit or as a test to judge the significance of an association between attributes. Chi – Square is calculated by the help of following formulae :

\chi ^{2}\; \left( cal \right)\; =\; \sum_{}^{}{\frac{\left( O-\mbox{E} \right)^{2}}{\mbox{E}}}


O = Observed frequency

E = Expected frequency

D.f = (r-1) (c-1)

Conditions for the application of Chi – square test :

  • Observation recorded and used are collected on the random basis.
  • All the items in the sample must be independent.
  • No – group should contain very few items.
  • The overall no if items must be greater than 30. i.e it should normally be at least 50.
  • The constraints  must be linear 

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