When to use a chi- square test?

The chi-square test can be used to test the in the following scenarios:
1. To test the Goodness of fit of the variables when their expected and observed frequencies are given.
2. To test the independence of the categorical variables by making it into a contingency table.
3. To test the significance of the single single variance with the given variance.
1). Goodness of fit:
Goodness of fit test is applied to check how well the sample data obtained fits the distribution of the selected population. It can also be viewed as whether the frequency distribution fits the given pattern. Most commonly used test to check the goodness of fit is the chi-square test.
There are two values involved are observed and the expected values. The observed value represents the frequency of particular category in the sample and the expected value is obtained from the given distribution. Moreover, it summarises the difference between the expected and observed values of the given data.
The hypotheses are stated as given below:
Null hypothesis: Data comes from the specified distribution.
Alternative hypothesis: Data does not come from the specified distribution.
The chi-square test statistic is calculated using the formula given below:
${\displaystyle {\chi }^2=\sum \frac{{(O_i-E_i)}^2}{E_i}}$
where,
${\displaystyle O_i}$ - Represents the observed value
${\displaystyle O{E}_i}$ - Represents the expected value
2). Test for independence:
In test for independence, we test whether there is an association between the categorical variables.
Null hypothesis: There is no association between the two categorical variables.
Alternative hypothesis: There is no association between the two categorical variables.
The chi-square test statistic is calculated using the formula given below:
${\displaystyle {\chi }^2=\sum \sum \frac{{(O_{ij}-E_{ij})}^2}{E_{ij}}}$
${\displaystyle O_{ij}}$ - Represents the observed values
in ${\displaystyle i^{th}}$ column and ${\displaystyle j^{th}}$ row
${\displaystyle E_{ij}}$ - Represents the expected value
in ${\displaystyle i^{th}}$ column and ${\displaystyle j^{th}}$ row
3). Test for single variance:
Here, the chi-square test is used to compare the single sample variance with the population variance.
Null hypothesis: The given sample variance is equal to the population variance.
Alternative hypothesis: The given sample variance is not equal (less or greater) to the population variance.
The chi-square test statistic is calculated using the formula given below:
${\displaystyle {\chi }^2=\frac{(n-1)s^2}{{\sigma }^2}}$
where,
n - Sample size
${\displaystyle s^2}$ - Sample variance
${\displaystyle {\sigma }^2}$ - Population variance