• The chi-square goodness of fit test and the chi-square test of independence test hypotheses about proportions.
  
  
• Chi-square tests examine whether the observed number of cases in a sample match what we expect in the population.
  
  
• Degrees of freedom for chi-square tests are based on the number of categories and not the number of cases.
  
  
• The chi-square goodness-of-fit test and the chi-square test of independence have the same general form but have different procedures for calculating the expected value (E) and the degrees of freedom.
  
  
• For many years, statisticians thought that we needed to worry about the validity of our chi-square test if E fell below 5-10. The current thinking is that we do not need to worry about our expected values.

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