• If you have count data, you can do a chi-square test.
• Examples
• Number of men vs. women
• Number of heart attacks in a drug study
• Comparing number of students attending two classes
• etc.

• Calculate the number of observations expected in each cell or condition (according to the null hypothesis).
• Compare how well the actual data corresponds to the null hypothesis.
• If the fit is poor, reject the null hypothesis

• Like the ANOVA or F test, it is one tailed (X² is always positive).
• Like the t-test, X² is different for different degrees of freedom

Here is the observed data:
 

A	B	C
10	10	40

Calculate the chi square statistic. (10-20)²/20+(10-20)²/20+(40-20)²/20=30

• This is the (observed-expected)²/expected for each of the three cells.
• A larger X² signals greater deviance from the null hypothesis.

Is this significant?  There are c-1=2 degrees of freedom. The critical value for X² is 5.99.



Like the t and F, different distributions for different degrees of freedom. Only the right end of the tail matters.

2 degrees of freedom
 
 
 
 
 
 

4 degrees of freedom



Chi-square tests can be done with two independent variables

• Same logic
• Null hypothesis is that there is no relationship between the rows and columns.
• degrees of freedom are (r-1)(c-1)
• It's a little trickier to calculate expected counts, but same idea.

estimated Chi-Square X²

• Sum (observed count - expected count)²/expected count
• expected cell count=(row total*columns total)/n
• (45*67)/134=22.5
• (45/134)*67=22.5

Here's some Data (Male/Female vs. 4 sport motivations):

	Female	Male	Total
Popularity	14	31	45
Fitness	7	18	25
Self-Confidence	21	5	26
Entertainment	25	13	38
Total	67	67	134

Null hypothesis is that there is no association between the rows and columns.

• Calculate X²
• degrees of freedom are (r-1)(c-1)= 3
• Use the table
• bigger  cell counts are better
• don't use when expected count is less than 5