# Chi-Square

Chi-Square is an important non-parametric test and as such, rigid assumptions are necessary regarding 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 enables us to explain how the theoretical distribution fits to the observed data. If the calculated value of the chi-square is less than the table value at a certain level of significance it means that there is no difference between observed and expected frequencies otherwise there is a significant difference between expected and observed frequencies.

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

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

Where O = Observed frequency

E = Expected frequency

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

Conditions for the application of the Chi-square test :

• Observations recorded and used are collected on a 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