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Chi-squared test: appropriate data

This is a test used to evaluate an hypothesis by comparing measured results to theoretically expected results (i.e. the Null hypothesis). This test is designed to convert the differences between “expected” and “measured” results into the probability of their occurring by chance. After using the conversion calculations, it basically compares a calculated value X2 with an expected value X2 (Chi-Square value) and tells the observer if the difference was “significant” enough to reject the expected result (null hypothesis). It takes into account both the sample size and the number of variables. As the number of variables increase, so does the complexity.

For example, suppose that the overall male to female ratio of UVA Anesthesiology Faculty was 1:1, but over the past 5 years the Critical Care/ Anesthesiology faculty had 8 males and 4 females. You might want to know if this would be considered a significant difference from the expected 1:1 ratio for the department?

To test this question, first build a table showing observed numbers (O), expected numbers (E). Then you subtract each “expected” value from the corresponding “observed” value (O-E), then square the “O-E” values, and divide each by the relevant “expected” value to give (O-E)2/ E. Add all the (O-E)2/E values and call the total “X2”.

See Calculating the X2 Value Table

Now you compare the X2 value (1.3334) with a X2 (chi squared) value in a standard table (see below) of X2 values and “degrees of freedom”. Degrees of freedom = n-1 (n= # of categories, in this case 2: male and female). Thus, our example has 1 degree of freedom.

See the Table of Standard X2 Values

Back to the original question of: “is the male: female ratio of 2:1 for Critical Care/ Anesthesiology faculty significant”? A significant difference from the 1:1 faculty ratio hypothesis (null hypothesis) would be found if the calculated X2 value was greater than the expected X2 value (3.84) shown for 0.05 column. If so, it would mean that there was a 95% chance of some unknown bias towards male faculty entering Critical Care/ Anesthesia and only a 5% probability that our calculated X2 value would have occurred by chance. Thus, finding a significant difference would give a reason to reject the null hypothesis of a 1:1 male to female ratio.

For our example, looking at the chart, for a p= 0.05, the X2 expected value is 3.84. Our X2 value (1.3334) was NOT greater than 3.84, thus the finding of 8 males and 4 female Critical Care/ Anesthesiology faculty would NOT be considered a significant departure from the 1:1 ratio in the entire department. Thus, the null hypothesis of a 1:1 ratio is still reasonably true.

To carry this further, if the calculated X2 value was equal to or less than the expected X2 for the p = 0.95 column, the results give no reason to reject your hypothesis that the faculty has a 1:1 ratio. Rarely, a calculated X2 value lower than the X2 value in the p =0.95 or 0.99 column gives evidence that the calculated results actually agree well with the “null” hypothesis.

On the other hand, if our calculated X2 value exceeded the value of 10.83 in the “p = 0.001” column, not only would it be significant, but it would have given us 99.9% confidence that some factor leads to a “bias” towards males entering the Critical Care/ Anesthesiology field at UVA, and only a 0.1% chance the difference was caused by chance.

In summary, the Chi-square test for categorical variables determines whether there is a difference in the population proportions between two or more groups.