1-Chi-square toggle

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Chi Sounds like “Hi” but with a K, so say Chi-Square like “Ki square”

And Chi is the Greek letter Χ, so we can also write it Χ2

Chi-square – One Way. Also known as Goodness of Fit test

Given a survey that asked the question: Which of these do you prefer as a pet? Cats, Dogs, Birds, Fish, Snakes.

Because the survey also recorded the sex of the respondent, these are possible research questions:

  1. Do men have a preference for Type of Pet?
  2. Do women have a preference for Type of Pet?
  3. Is the Type of Pet preference for women in this survey different from that of men in previous surveys?
  4. Is the Type of Pet preference related to Sex of the respondent?

Questions 1, 2, and 3 can be answered using a One-way Chi-Square test. Question 4 can be answered by using a Two-way Chi-square Here.

Note: although questions 3 and 4 are similar, they are different questions. Question 3 is about a sample distribution compared to a known or assumed distribution. Question 4 is about the categorical variables Sex and Type of Pet being related (not independent). To answer Question 3, we use the historical distribution as the “expected” distribution.

The one-way Chi-square test determines if there is a statistically significant difference in an actual frequency distribution from the expected distribution.

Questions 1 and 2 can be answered using a “uniform” expected distribution, i.e. the total number of responses to the survey is equally distributed among the three categories of pets. But there could be a different expected distribution.


A radio station claims that the listening audience in its broadcast region has this distribution of music preferences.

A marketing executive surveys 500 radio listeners in the broadcast region and gets these results:

Question: Does the distribution of the survey results differ from the claimed distribution?

The Test for Homogeneity is used when there are two populations or subgroups and only one categorical variable.

Canada has universal health care. The United States does not but often offers more elaborate treatment to patients with access. How do the two systems compare in treating heart attacks? Researchers compared random samples of U.S. and Canadian heart attack patients. One key outcome was the patients’ own assessment of their quality of life relative to what it had been before the heart attack. Here are the data for the patients who survived a year:    


Is there a significant difference between the two distributions of quality-of-life ratings?

This is a Test for Homogeneity because there are two populations, Canada and US, and only one categorical variable, Quality of Life.


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