3. The computation of chi square

In "What Works: Effective Recidivism Reduction and Risk-Focused Prevention Programs," Roger Przybylski compiles evidence-based options for preventing new and persistent criminal behavior. You are particularly interested in the idea that substance abuse programs can reduce recidivism. [Source:Roger Przybylski. 2008. What Works: Effective Recidivism Reduction and Risk-Focused Prevention Programs.Denver, CO: Colorado Division of Criminal Justice. Retrieved from http://dcj.state.co.us/ors/pdf/docs/WW08_022808.pdf.]

Suppose you conduct a study using a sample of 300 offenders released from different prisons to see whether participation in a substance abuse program affects how likely they are to commit subsequent crimes.

The following table shows the results of your study:

Observed Frequencies for Recidivism and Substance Abuse Programs

	Substance Abuse Program
Recidivism	Yes	No	Totals
Reoffends	63	74	137
Does not reoffend	96	67	163
Totals	159	141	300

The null hypothesis is that there is no relationship between participation in a substance abuse program and recidivism. If these variables were independent, the cell frequencies would be determined by random chance. The chi square test statistic measures the contribution of the categorization above and beyond what is expected by chance. The formula for the chi square statistic is:

2(obtained)=(fofe)2fe

Since there are several steps required to compute this statistic, it is helpful to use a computing table to organize the results of each step.

Computational Table for Recidivism and Substance Abuse Programs

63
74
96
67
N = 300	N =	N =		2

(obtained) =

Step 1.Calculate the expected frequencies for each of the four variable combinations and enter them into the second column of the table using the dropdown menus. Then, sum the expected frequencies and choose the correct value for the sum in the dropdown menu at the bottom of the fe column.

Note:The table lists the observed frequencies (fo) in column 1 in order from the upper-left-hand cell to the lower-right-hand cell, moving left to right and top to bottom across the table. Column 2 lists the expected frequencies (fe) in exactly the same order. Double-check to make sure you have listed the cell frequencies in the same order for both of these columns.

Step 2.Calculate the difference between the observed and the expected frequencies for each of the four variable combinations, as well as the sum of these calculations, and then enter them into the third column of the table using the dropdown menus.

Step 3. Calculate the square of the difference between the observed and the expected frequencies for each of the four variable combinations, and then enter them into the fourth column of the table using the dropdown menus.

Step 4.Divide the square of the difference between the observed and the expected frequencies by the expected frequency for each of the four variable combinations, and then enter them into the fifth column of the table using the dropdown menus.

Step 5.Finally, add up column 5 and enter the value into the bottom cell of the fifth column of the table using the dropdown menu.

Now that you have calculated the chi square test statistic, you will need to compare the test statistic with the critical region and interpret the results.