DIGSTATS: Chi-Squared

Chi-Squared Tests - c²

Chi-Squared for Goodness of Fit

Chi-Squared for goodness of fit uses a set of predicted data to determine how close the predictions were to the observed frequencies. A statistic c² (the Greek letter chi, pronounced "kye", squared) is calculated that represents the degree to which each observed frequency in a category differs from the expected frequency for that category. We decide to set the expected values so that an equal part of the population fits into each category. For example, if someone were testing 36 children to see whether they choose to eat M&Ms, Cotton Candy, or Snickers, then they would expect that 12 children (36 children divided into 3 categories) would pick each food. So if someone were to actually run this experiment, the might end up with these data:

	M&M	Cotton Candy	Snickers
Observed	19	8	9
Expected	12	12	12
Warning! The above data were falsified and should not be taken as genuine.

So the null hypothesis is that the experimenter's values predict closely the observed values. The alternative hypothesis is that they do not. These hypotheses hold for all Chi-Squared goodness of fit tests. Since the expected data in the above experiment were distributed evenly to each of the three candies, the hypotheses for this test are the following:

H_O: M&M, Snickers, and Cotton Candy appear with equal frequency.
H_A: M&M, Snickers, and Cotton Candy do not appear with equal frequency.

To complete the test, the chi-square and df values must be calculated. In order to ease this fairly formidable process, DIG Stats provides the use of a Chi-Squared goodness of fit calculator which can be run in most recent web browsers. Click the button below to see if your browser can run the applet.

With our chi-square and df in hand, we find the significance of our results by using a Chi-Squared table (if that table is not accurate enough, try this one). To decide whether to reject our null hypothesis, we can look up the "critical Chi-Squared" value on the table that corresponds to the df and the selected alpha level (usually a=.05 or a=.01). If the calculated Chi-Squared is greater than the value in the table, then we reject null hypothesis and conclude that the predictions we were making were incorrect.

The applet uses a slightly different, but equivalent method for determining the significance of results. Instead of using a Chi-Squared table, it calculates the alpha so that if you were to look up the calculated df and the calculated alpha in a table, the corresponding critical chi-square would be the calculated Chi-Squared. This calculated alpha is called the p-value. This can be used to test the hypotheses by rejecting the null hypothesis if the p-value is lower than the alpha you selected.

The Cellular Phone and Dice activities in the previous menu use goodness of fit tests.

Original work on this document was done by Central Virginia Governor's School students Richard Barnes, Kim Tibbs, and Ryan Nash (Class of '00).

This document was updated by Central Virginia Governor's School students Matthew James and Kyle Nenninger
(Class of '03).