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)
