The chi-square test is a non-parametric test that is used in statistic to determine if an observed distribution or model conforms or is similar to an expected distribution or model. In simple terms, this test will tell you if the data you collected is similar to other data or to what you expected.
There are several types of chi-square test such as the Chi-square Test of Independence, which is used for nominal data, and the Goodness-of-Fit Test, 
A unique caveat of chi-square test is that we normally desire as a researcher to make sure we do not reject our model. This is opposite of traditional hypothesis testing which desires often to reject the null hypothesis as this indicates that there is a statistical difference. With chi-square test, we want our observed model to be similar to the values found in the expected model. What this means is that our model represents what is happening in the real-world and is not only theoretical. If we reject the null it means that the model we are trying to create is not similar to expected values that might be found in the real world. In other words, we found something that does not conform to what is expected. If a model does not represent the world, it may not serve much purpose.
Here are the assumptions of Goodness-of-Fit Test
- Random selection of subjects
- Mutually exclusive categories
Here are the steps
- Determine hypothesis
- H0: There is no difference between the observed values/model and the expected values/model
- H1: There is a difference between the observed values/model and the expected values/model
- Decide level of significance
- Determine degree of freedom to find chi-square critical
- Compute for the expected frequencies
- Compute chi-square
- Make decision to accept or reject null
- State conclusion
Here is an example
A principal wants to know if the number of students absent each day of the week is the same. Below are the results for one week.
Day Absents
Monday 17
Tuesday 20
Wednesday 16
Thursday 14
Friday 13
Step 1: Determine Hypothesis
- H0: The number of students absent is the same every day
- H1: The number of students absent is not the same every day
Step 2: Decide level of significance
- 0.05
Step 3 Determine chi-square critical region (computer does this for you)
- Chi-square critical region = 9.48
Step 4: Compute expected frequencies
- Computer does this
Step 5: Compute Chi square (computer does this for you)
- Chi-square = 1.87
Step 6: Make decision
- Since the computed chi-square of 1,87 is less than the critical chi-square value of 9.48 we do not reject the null hypothesis
Step 7: Conclusion
- Since we do not reject the null hypothesis we can say that there is a lack of evidence that there is a difference in the number of absences each day of the week. In other words, the number of students absent each day is the same.
NOTE: There is also a way to do this test when the expected frequencies are unequal
parametric test such as the data being normally distributed. This allows a researcher to still make inferences from data that may not have normality. In addition, non-parametric test are used for data that is at the ordinal or nominal level. In many ways, Spearman correlation and Pearson product moment correlation compliment each other. One is used in non-parametric statistics and the other for parametric statistics and each analyzes the relationship between variables.