U8.7 Choosing Categorical Procedures
Learn to pick the right categorical inference procedure on the AP exam: 1-prop z, 2-prop z, chi-square GOF, homogeneity, or independence.
What you'll do in this lesson
A voice-first session with the Crimsora tutor on U8.7 Choosing Categorical Procedures, then targeted practice and FRQs — with the tutor adapting to where you get stuck.
What this lesson covers
By Unit 8.7 you already know how to run each categorical procedure. The hard part on the AP exam is not the arithmetic — it's reading a scenario and deciding which test belongs there. A single wrong choice can cost you every point on an FRQ, even if your mechanics are flawless.
This lesson gives you a reliable decision process. You will learn to count how many variables and how many samples are in play, whether your data are counts or proportions, and how the sampling design (one sample split by two variables versus several separate samples) tells homogeneity apart from independence. Master these signals and the choice becomes automatic.
This lesson gives you a reliable decision process. You will learn to count how many variables and how many samples are in play, whether your data are counts or proportions, and how the sampling design (one sample split by two variables versus several separate samples) tells homogeneity apart from independence. Master these signals and the choice becomes automatic.
The Five Procedures at a Glance
Every categorical inference question on the AP exam maps to one of five procedures. The fastest way to choose is to ask two questions: How many categorical variables are being measured, and how many samples (populations) are involved?
Notice the z-procedures handle exactly two outcomes (success/failure), while chi-square procedures handle two or more categories. When a variable has only two categories and you compare two groups, both 2-prop and chi-square homogeneity technically apply — but if the question asks about a difference in proportions or a direction, use 2-prop ; if it asks whether distributions are the same, chi-square works. The AP exam usually signals which by the way the claim is phrased.
| Procedure | Variables | Samples | Typical question |
|---|---|---|---|
| 1-prop | 1 (two outcomes) | 1 | Is a single proportion equal to a claimed value? |
| 2-prop | 1 (two outcomes) | 2 | Do two groups differ in one proportion? |
| Chi-square GOF | 1 (≥2 categories) | 1 | Does one distribution match a claimed model? |
| Chi-square homogeneity | 1 | ≥2 | Do several groups share the same distribution? |
| Chi-square independence | 2 | 1 | Are two variables associated in one population? |
Counts of Categories vs. Comparing Proportions
The first fork in the road: are you working with a proportion (a single success/failure percentage) or with a full distribution across several categories?
Use a z-procedure when the variable is binary and the question is about the proportion of successes. Key phrases include "the proportion who...", "more than half", "the percentage that improved", or an explicit claimed value like . One sample points to 1-prop ; two independent samples point to 2-prop .
Use a chi-square procedure when the variable has three or more categories, OR when you are checking whether an entire distribution fits a model. If someone rolls a die and asks whether all six faces are equally likely, that is a goodness-of-fit test — six categories, one sample, one claimed model. A common misconception is reaching for chi-square whenever you see a table. Instead, ask what the table represents. A table showing counts in categories for a single group compared to expected proportions is GOF; a two-way table cross-classifying subjects is homogeneity or independence.
Remember: chi-square tests never tell you direction. They only detect whether observed counts differ from what a model predicts. If a question asks "is proportion A greater than proportion B," chi-square cannot answer that — you need a 2-prop test.
Use a z-procedure when the variable is binary and the question is about the proportion of successes. Key phrases include "the proportion who...", "more than half", "the percentage that improved", or an explicit claimed value like . One sample points to 1-prop ; two independent samples point to 2-prop .
Use a chi-square procedure when the variable has three or more categories, OR when you are checking whether an entire distribution fits a model. If someone rolls a die and asks whether all six faces are equally likely, that is a goodness-of-fit test — six categories, one sample, one claimed model. A common misconception is reaching for chi-square whenever you see a table. Instead, ask what the table represents. A table showing counts in categories for a single group compared to expected proportions is GOF; a two-way table cross-classifying subjects is homogeneity or independence.
Remember: chi-square tests never tell you direction. They only detect whether observed counts differ from what a model predicts. If a question asks "is proportion A greater than proportion B," chi-square cannot answer that — you need a 2-prop test.
Homogeneity vs. Independence: It's About Sampling Design
The trickiest distinction in this unit is chi-square homogeneity versus chi-square independence, because the mechanics (expected counts, test statistic, degrees of freedom) are identical. The difference lives entirely in how the data were collected.
Chi-square homogeneity applies when you take separate samples from two or more populations or groups (or assign subjects to treatments) and measure one categorical variable. The group sizes are usually fixed by the researcher in advance. Example: survey 100 freshmen, 100 sophomores, and 100 juniors and record favorite cafeteria meal. You chose the group totals, so you are comparing distributions across groups.
Chi-square independence applies when you take one single sample and classify each individual on two categorical variables. The totals in the margins are all random, not fixed. Example: survey 300 randomly chosen students and record both class year and favorite meal. You want to know whether the two variables are associated.
On the AP exam, read the data-collection sentence carefully: multiple samples means homogeneity; one sample split two ways means independence.
Chi-square homogeneity applies when you take separate samples from two or more populations or groups (or assign subjects to treatments) and measure one categorical variable. The group sizes are usually fixed by the researcher in advance. Example: survey 100 freshmen, 100 sophomores, and 100 juniors and record favorite cafeteria meal. You chose the group totals, so you are comparing distributions across groups.
Chi-square independence applies when you take one single sample and classify each individual on two categorical variables. The totals in the margins are all random, not fixed. Example: survey 300 randomly chosen students and record both class year and favorite meal. You want to know whether the two variables are associated.
| Feature | Homogeneity | Independence |
|---|---|---|
| Number of samples | 2+ | 1 |
| Number of variables measured | 1 | 2 |
| Margins fixed by design | Yes (group sizes) | No |
| Conclusion wording | "same distribution across groups" | "association between two variables" |
A Decision Checklist and How the Exam Tests It
Turn choosing into a routine. Work through these questions in order.
First, is the data categorical? If it is means or quantities, you need a t-procedure instead — not covered here. Second, how many categorical variables? Two variables from one sample points to independence. Third, if one variable, how many groups or samples? One sample checking a claimed distribution across 3+ categories is GOF; a claimed single proportion is 1-prop . Fourth, if you have multiple samples: one binary variable across two groups can be 2-prop (for a difference or direction) or chi-square homogeneity (for identical distributions); three or more categories or three or more groups forces chi-square homogeneity.
The exam tests this in two ways. Multiple-choice items give a short scenario and ask which procedure is appropriate — read for sample count and variable count. Free-response items ask you to "identify the appropriate procedure and justify," so you must name it AND state the reason ("one sample classified on two variables, so chi-square test for independence"). Always state the name precisely; writing "chi-square test" without specifying homogeneity or independence often loses credit. Then check conditions: random sampling/assignment, independence (10% rule), and the Large Counts condition (all expected counts for chi-square, or and for z).
First, is the data categorical? If it is means or quantities, you need a t-procedure instead — not covered here. Second, how many categorical variables? Two variables from one sample points to independence. Third, if one variable, how many groups or samples? One sample checking a claimed distribution across 3+ categories is GOF; a claimed single proportion is 1-prop . Fourth, if you have multiple samples: one binary variable across two groups can be 2-prop (for a difference or direction) or chi-square homogeneity (for identical distributions); three or more categories or three or more groups forces chi-square homogeneity.
The exam tests this in two ways. Multiple-choice items give a short scenario and ask which procedure is appropriate — read for sample count and variable count. Free-response items ask you to "identify the appropriate procedure and justify," so you must name it AND state the reason ("one sample classified on two variables, so chi-square test for independence"). Always state the name precisely; writing "chi-square test" without specifying homogeneity or independence often loses credit. Then check conditions: random sampling/assignment, independence (10% rule), and the Large Counts condition (all expected counts for chi-square, or and for z).
Key terms
- 1-proportion z-test.
- Inference for a single population proportion compared to a claimed value ; requires one sample and a binary variable.
- 2-proportion z-test.
- Inference comparing proportions from two independent samples; used when a question asks about the difference or direction between two groups' proportions.
- Chi-square goodness-of-fit test.
- Tests whether one sample's distribution across three or more categories matches a claimed or expected distribution.
- Chi-square test for homogeneity.
- Tests whether the distribution of one categorical variable is the same across two or more separate samples or groups.
- Chi-square test for independence.
- Tests whether two categorical variables are associated within a single sample classified on both variables.
- Large Counts condition.
- For chi-square tests, all expected counts must be at least 5; for z-procedures, expected successes and failures must be at least 10.
- Two-way table.
- A table cross-classifying observations by two categorical variables (or by group and one variable), used for homogeneity and independence tests.
Worked example
A researcher randomly selects 250 adults and records each person's blood type (O, A, B, AB) and whether they have a certain allergy (yes/no). She wants to know if blood type and allergy status are related. Which inference procedure should she use, and why?
Start with the decision checklist. Are the data categorical? Yes — blood type has four categories and allergy status has two, both categorical.
How many categorical variables? Two: blood type and allergy status. That already narrows us toward a chi-square test for a two-way table rather than any z-procedure or GOF.
Now separate homogeneity from independence by looking at sampling design. The researcher took one random sample of 250 adults and then classified each person on both variables. She did not draw separate samples of, say, allergic and non-allergic people with fixed sizes. Because there is a single sample classified two ways, this is a chi-square test for independence.
State it precisely: use a chi-square test for independence to determine whether blood type and allergy status are associated in the population of adults. Justify by noting one sample, two categorical variables measured on each individual.
Before computing, confirm conditions: the sample was random; 250 is presumably less than 10% of all adults; and all expected counts should be checked to be at least 5. If a low-frequency cell like AB fails Large Counts, categories may need to be combined.
How many categorical variables? Two: blood type and allergy status. That already narrows us toward a chi-square test for a two-way table rather than any z-procedure or GOF.
Now separate homogeneity from independence by looking at sampling design. The researcher took one random sample of 250 adults and then classified each person on both variables. She did not draw separate samples of, say, allergic and non-allergic people with fixed sizes. Because there is a single sample classified two ways, this is a chi-square test for independence.
State it precisely: use a chi-square test for independence to determine whether blood type and allergy status are associated in the population of adults. Justify by noting one sample, two categorical variables measured on each individual.
Before computing, confirm conditions: the sample was random; 250 is presumably less than 10% of all adults; and all expected counts should be checked to be at least 5. If a low-frequency cell like AB fails Large Counts, categories may need to be combined.
Practice questions
A school offers three lunch periods. The principal randomly samples 80 students from each period and records whether each student buys lunch or brings lunch. She wants to know if the proportion buying lunch is the same across all three periods. Which procedure is appropriate?
- Chi-square goodness-of-fit test
- Chi-square test for homogeneity
- Chi-square test for independence
- 2-proportion z-test
Answer: Chi-square test for homogeneity
There are three separate samples (one from each lunch period), each of fixed size 80, and one categorical variable (buy or bring). Comparing the distribution of one variable across multiple predetermined samples is a test for homogeneity. It is not independence because the data come from three fixed samples, not one sample classified two ways. It is not a 2-prop z-test because there are three groups, not two.
A quality inspector claims that a bag of candy should contain colors in the ratio 3:2:2:1:2 across five colors. She opens one bag, counts the number of each color, and wants to test the claim. Name the appropriate procedure and briefly justify your choice.
Answer: Chi-square goodness-of-fit test.
There is one sample (one bag) and one categorical variable (color) with five categories, and the question compares the observed counts to a single claimed distribution. That is exactly the setup for a goodness-of-fit test. A z-procedure is wrong because the variable has more than two categories, and a two-way table test is wrong because only one variable is measured against a claimed model rather than two variables being cross-classified.
A researcher wants to know whether a larger proportion of people who exercise regularly report good sleep compared to people who do not exercise. She surveys two independent groups. Which procedure best answers her directional question?
- Chi-square test for homogeneity
- Chi-square test for independence
- 2-proportion z-test
- 1-proportion z-test
Answer: 2-proportion z-test
Two independent samples (exercisers and non-exercisers), one binary variable (good sleep or not), and a directional question about whether one proportion is larger point to a 2-proportion z-test. Chi-square homogeneity could compare the distributions but cannot address direction, and the question specifically asks whether one proportion is greater, which requires the z-procedure's one-sided alternative.
FAQ
- How do I tell the difference between chi-square homogeneity and independence?
- Look at how the data were collected. If the researcher drew two or more separate samples (or assigned subjects to groups) and measured one variable, it is homogeneity. If the researcher drew one single sample and classified each individual on two variables, it is independence. The calculations are identical, but the AP exam expects the correct name based on sampling design.
- When should I use a 2-prop z-test instead of a chi-square test?
- Use a 2-prop z-test when you have exactly two groups, a binary variable, and you care about the difference or direction between the two proportions (for example, whether one is larger). Use chi-square when there are more than two groups or categories, or when the question is simply whether the distributions are the same, since chi-square cannot detect direction.
- Does a two-way table always mean a chi-square test?
- A genuine two-way table that cross-classifies subjects points to homogeneity or independence. But a table listing observed versus expected counts for one variable's categories in a single sample is a goodness-of-fit setup. Read what the table represents rather than reacting to its shape.
- On the free-response section, is it enough to write 'chi-square test'?
- No. You should name the specific procedure — goodness-of-fit, homogeneity, or independence — and justify it by referencing the number of samples and variables. Writing only 'chi-square test' often loses credit because it does not show you understand the sampling design distinction.
Learn this with a teacher, not a page
The Crimsora tutor teaches U8.7 Choosing Categorical Procedures live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.