Chi-Square Test James A. Pershing, Ph.D. Indiana University

Basic Use of Chi-Square Examine the effects of an independent variable on the dependent variable Data are nominal (labels or names) What is collected is frequency data Can have any number of independent categories Data are presented in table format

Chi-Square -- Two Applications  Goodness-of-fit test to assess if a distribution fits (matches) a theoretical distribution  Test statistic to assess the statistical significance of a finding

Chi-Square as a Goodness-of-Fit Test Does a pattern of frequencies of a single categorical variable significantly differ from an expected pattern We choose the expected pattern Example: –Car colors of red -- blue -- black -- white –100 people given four pictures of same car in four different colors –If no preference -- expect 1/4 or 25 for each color

Car Color Preference Assumption = No Preference – Choices Are Random RedBlueBlackWhite Cars – Four Colors Expected Values

Car Color Preference Assumption = No Preference – Choices Are Random RedBlueBlackWhite Cars – Four Colors Observed Values

Car Color Preference Assumption = No Preference – Choices Are Random RedBlueBlackWhite Cars – Four Colors Color Expected Observed

Car Color Preference Assumption = No Preference – Choices Are Random RedBlueBlackWhite – 25 = = = = Cars – Four Colors Color Expected Observed Difference O - E

Calculating The Chi-Square Chi-Square Formula 22 =  (O – E) 2 E = (+23) 2 (-10) 2 (-15) 2 (+2) = = = 34.32

Chi-Square Goodness-Of-Fit Example  2 = Degrees of Freedom (DF) Number of categories minus one! DF = C car colors = 3

Critical Values of the Chi-Square Distribution at 5% Level The calculated value of the  2 must be larger than or equal to the table value for significance. Car color value = 34.32, so significant. df Critical value for 

Car Color Preference Assumption = No Preference – Choices Are Random RedBlueBlackWhite Color Expected Observed Because our observed value (34.32) for the observed differences is greater than the critical value (7.81) we can conclude that the differences are statistically significant at the.05 level.

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