CHI-SQUARE TEST

The chi-square (I) test is used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories. Do the number of individuals or objects that fall in each category differ significantly from the number you would expect? Is this difference between the expected and observed due to sampling error, or is it a real difference?

 Chi-Square Test Requirements

1. Quantitative data.

2. One or more categories.

3. Independent observations.

4. Adequate sample size (at least 10).

5. Simple random sample.

6. Data in frequency form.

7. All observations must be used.

 Expected Frequencies

When you find the value for chi square, you determine whether the observed frequencies differ significantly from the expected frequencies. You find the expected frequencies for chi square in three ways:

 I . You hypothesize that all the frequencies are equal in each category. For example, you might expect that half of the entering freshmen class of 200 at Tech College will be identified as women and half as men. You figure the expected frequency by dividing the number in the sample by the number of categories. In this exam pie, where there are 200 entering freshmen and two categories, male and female, you divide your sample of  200 by 2, the number of categories, to get 100 (expected frequencies) in each category.

 2. You determine the expected frequencies on the basis of some prior knowledge. Let's use the Tech College example again, but this time pretend we have prior knowledge of the frequencies of men and women in each category from last year's entering class, when 60% of the freshmen were men and 40% were women. This year you might expect that 60% of the total would be men and 40% would be women. You find the expected frequencies by multiplying the sample size by each of the hypothesized population proportions. If the freshmen total were 200, you would expect 120 to be men (60% x 200) and 80 to be women (40% x 200).

 Now let's take a situation, find the expected frequencies, and use the chi-square test to solve the problem. Degrees of freedom (df) refers to the number of values that are free to vary after restriction has been placed on the data. For instance, if you have four numbers with the restriction that their sum has to be 50, then three of these numbers can be anything, they are free to vary, but the fourth number definitely is restricted. For example, the first three numbers could be 15, 20, and 5, adding up to 40; then the fourth number has to be 10 in order that they sum to 50. The degrees of freedom for these values are then three.The degrees of freedom here is defined as N - 1, the number in the group minus one restriction (4 - I ).

 3. Find the table value for Chi Square. Begin by finding the df found in step 2 along the left hand side of the table. Run your fingers across the proper row until you reach the predetermined level of significance (.05) at the column heading on the top of the table. The table value for Chi Square in the correct box of 4 df and P=.05 level of significance is 9.49.

 4. If the calculated chi-square value for the set of data you are analyzing (26.95) is equal to or greater than the table value (9.49 ), reject the null hypothesis. There IS a significant difference between the data sets that cannot be due to chance alone. If the number you calculate is LESS than the number you find on the table, than you can probably say that any differences are due to chance alone.

 In this situation, the rejection of the null hypothesis means that the differences between the expected frequencies (based upon last year's car sales) and the observed frequencies (based upon this year's poll taken by Thai) are not due to chance. That is, they are not due to chance variation in the sample Thai took; there is a real difference between them. Therefore, in deciding what color autos to stock, it would be to Thai's advantage to pay careful attention to the results of her poll!

The steps in using the chi-square test may be summarized as follows:

 Chi-Square I. Write the observed frequencies in column O

Test Summary 2. Figure the expected frequencies and write them in column E.

3. Use the formula to find the chi-square value:

4. Find the df. (N-1)

5. Find the table value (consult the Chi Square Table.)

6. If your chi-square value is equal to or greater than the table value, reject the null

hypothesis: differences in your data are not due to chance alone

For example, the reason observed frequencies in a fruit fly genetic breeding lab did not match expected

frequencies could be due to such influences as:

• Mate selection (certain flies may prefer certain mates)

• Too small of a sample size was used

• Incorrect identification of male or female flies

• The wrong genetic cross was sent from the lab

• The flies were mixed in the bottle (carrying unexpected alleles)