Lab #7: Chi-Square. In case you re curious, the study was published in Here s the reference:

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1 Psychology 210 Lab #7: Chi-Square As usual, log in and launch SPSS and open up a data file. The data set of the day is called michigankids.sav and is located on the class web page. Open it up and we ll get started The data come from a study of grade school children in Michigan. As you can see, the file contains an array of demographic variables, along with children s responses to the following question: What would you most like to do at school? A. Make good grades B. Be good at sports C. Be popular In case you re curious, the study was published in Here s the reference: Chase, M. A., and Dummer, G. M. (1992), "The Role of Sports as a Social Determinant for Children," Research Quarterly for Exercise and Sport, 63, Today we ll be continuing along the theme of categorical (a.k.a. nominal) data analysis with two versions of the Chi-square test. Let s start with the simpler of the two: the Chi-square goodness of fit test. This is very similar to the binomial (in some cases, identical), but isn t limited to variables that have only two possible values. Therefore one could analyze variables like political party, country of citizenship or freshman residence hall that have more than two options and therefore wouldn t lend themselves to a binomial. To run a Chi-square, select Analyze Nonparametric Tests Chi-Square. A window like the following should appear: (I bet you didn t see that coming) Select a variable of interest and send it to the Test list like you ve done before. Let s use the critical variable mentioned above (goal). Remember, the chi-square will determine if we have an even mix of each response, or if children were more likely to choose one response over the

2 rest. In most cases (including this one) you ll want to test an equal distribution of expected values. Note however that I can test a different set of expected values by entering them manually in the lower-right compartment. For instance, if I were testing political affiliations, I might not want to assume that democrat and independent were equally likely on a liberal college campus. Anyway, when you re ready, click OK and wait for the output, which should look remarkably like this: The upper table shows observed and expected frequencies, along with the differences between the two (residual). Note that you could use these to calculate Chi-square by hand using the formula: = 2 ( O E) χ E Observed n = O Expected n = E Residual = (O-E) 2

3 If you were to do so, it should exactly match the chi-square value in the lower box. Also reported in the lower box are the degrees of freedom and significance level. Notice that I get a more precise estimate of probability by using SPSS than doing the analysis by hand and turning to a look-up table. With SPSS, I have the actual probability to 3 decimal places, and not simply whether it is greater than or less than.05. In this case, the chi-squared value, along with the reported frequencies tell me that kids in Michigan are more likely to value grades than the alternatives of sports and popularity. This, of course, is very reassuring. If I wanted to write about this like a real scientific-type, I d want to use this syntax: Grades, populatity, and sports were not equally popular among the sample, χ 2 (2,N=478)=80.515, p<.001 Where 2 is the degrees of freedom and N=248 reflects my total sample size. Chi-square Test for Independence You ll notice that the previous test (Chi-square goodness of fit) is limited to a single categorical variable. There are times when one would want to look at the relationship between two variables. If both variables are categorical, this calls for the slightly different Chi-square test for independence. So return to your data file, light some fireworks and hum a few bars of This Land is Your Land. To run the chi-square test for independence, select Analyze Descriptive Statistics Crosstabs. The following window will appear: Let s find out if students goals differ depending on their setting (whether the school is rural, urban or suburban). Select goals and setting as your row and column variables, and check the box marked Display clustered bar charts, just to be fancy. Check out the options that are available by clicking the statistics, cells and format buttons at the bottom of this window. Some of these won t look at all familiar and should be ignored for now.

4 Under the statistics sub-window, select Chi-square (top-left corner) to request a chi-square test for independence. Under the Cells sub-window, request observed and expected counts (upper left) as well as standardized and unstandardized residuals (bottom right). When you re ready, click OK Take a moment to scan the new tables on your output window: 1) Case processing summary. Ok, this part s pretty boring. Nothing to see here Move along 2) Crosstabulation. This displays the guts of the chi-square calculation. You should see everything you d need to calculate the statistic by hand, thanks to the requested options. Take a moment to figure out what a residual is in the chi-square formula. 3) Chi-square tests. The top line is the critical one. It shows a chi-square value, the number of degrees of freedom, and the probability of getting this distribution based on chance variation. Notice again that the probability is exact to 3 decimals, rather than the crude greater than or less than.05 you get from the chi-square table in your textbook. 4) Clustered bar chart. Ooooh pretty. In this case, the low probability tells me that the different preferences for students in different settings did not come about as a result of chance that is, students in different settings actually tend to have different goals. The bar chart at the bottom of your output helps you decode where the differences lie. Notice that the pattern for urban and suburban students is pretty similar, with the majority preferring good grades followed by popularity and then athletic success. Rural kids on the other hand, are more evenly distributed between the three goals. The format for writing about this test would be very similar to the previous example: goals and school setting were not independent of one another, χ 2 (4,N=478)=18.828, p<.002

5 To Summarize for future reference: Chi-Square Goodness of Fit: Analyze Nonparametric Tests Chi-Square Chi-Square Test for Independence: Analyze Descriptives Crosstabs Lab assignment: The superintendent of a Michigan school district returns from an education conference and asks you to run some analyses for her: 1. She has learned at her conference that boys are more likely to go away to a private school. Because of this, she is worried that uneven sex distributions in grade school classrooms might threaten normal social development. Based on the data here, does she have reason to worry about having a skewed ratio of boys to girls in Michigan schools? Present her with statistical evidence for your conclusion. 2. In her school district, many after school activities are single-sex. In order to properly allocate funding, she wants to know whether boys and girls are equally interested in academic, social, and athletic activities, or if there are differences in interest based on sex. Should the budget for girls clubs be allocated the same way as the budget for boys clubs? Again, present a statistical argument. Remember to interpret your results these are real kids! Based on what you know about sex differences and grade school children, does the outcome make sense?

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