Why Check the 10% Condition?
The previous tutorial, “Random Condition for Chi-Square Procedures,” focused on how a study’s randomness supports inference. A related question is whether the observations can reasonably be treated as independent. When people or other units are sampled from a finite population without replacement, selecting one unit changes which units remain available for later selections. The selections are therefore dependent in a strict mathematical sense.
For many AP Statistics procedures, including chi-square procedures, a sample that is small compared with its population allows us to treat those observations as approximately independent. The 10% condition is the usual check: when sampling without replacement, the sample size should be no more than 10% of the population size. This is a practical condition for the dependence created by sampling—not a test of whether the categorical variables themselves are related.
This condition does not replace the random condition. A sample can be less than 10% of a population and still be a voluntary-response sample rather than a random sample. Nor does the condition establish that a chi-square procedure’s expected counts are large enough. As discussed in earlier tutorials, those are separate checks.
How to Check the Condition
Let \(n\) be the number of individuals in the sample and \(N\) the size of the population from which the sample was drawn. The direct check compares \(n\) with 10% of \(N\). The equivalent check compares \(N\) with ten times \(n\).
For example, if a random sample contains 180 people, the population should contain at least \(10(180)=1{,}800\) people. If the population size is not stated, do not invent it or claim the 10% condition is met. Say that the condition cannot be verified from the information given.
Use the sample size of individuals, not the number of categories in a table, the number of cells, or a cell count. In a test of independence, \(n\) is the total number of sampled individuals classified by both variables. In a test of homogeneity based on separate random samples, check each sample against the size of its own population.
Determine whether the individuals were sampled without replacement from a finite population. If the study description does not say, do not assume details that are not given.
Count all sampled individuals in the relevant sample and identify the population from which they were selected.
Check whether \(n \le 0.10N\), or equivalently whether \(N \ge 10n\).
If met, the sample is small enough relative to the population to treat the observations as approximately independent. Continue checking the other conditions separately.
What the Condition Means for Chi-Square Tests
For a chi-square test of independence, one random sample is classified according to two categorical variables. The 10% condition concerns the sampled individuals: was the sample small enough relative to the population that selecting one individual does not substantially affect the chance of selecting another? It does not mean that the two categorical variables must be independent. That independence is the claim in the null hypothesis, as explained in “Stating Hypotheses for a Test of Independence.”
For a chi-square test of homogeneity, researchers may take separate random samples from several populations. If those samples are drawn without replacement, check the 10% condition separately for each sample and its corresponding population. A large population for one group does not make up for a sample that is too large relative to another group’s population.
A test of homogeneity may also arise from a randomized experiment. Random assignment is not the same as sampling a fixed percentage of a population. The 10% condition applies when the study uses sampling without replacement from a finite population; it is not automatically a requirement just because participants were randomly assigned to treatments. The design must still support treating experimental units as independent—for example, one unit’s treatment or response should not determine another unit’s response.
Worked Example: One Random Sample for Independence
Worked Example: One Random Sample for Independence
A fictional county selects a random sample of 240 residents without replacement from a county population of 12,000. Each resident reports both their usual way of commuting and whether they usually work a daytime or evening schedule. Researchers plan to use a chi-square test of independence to investigate an association between commute method and work schedule.
Identify the quantities. There is one sample, and every sampled resident is classified by both categorical variables. Thus \(n=240\) and \(N=12{,}000\). The population size is the number of county residents, not the number of possible table cells.
Check the 10% condition. Ten percent of the population is
Because \(240 \le 1{,}200\), the sample is no more than 10% of the population. Equivalently, \(10n=10(240)=2{,}400\), and \(12{,}000 \ge 2{,}400\). Both checks show that the condition is met.
Interpret the check. Because the sample is small relative to the county population, the residents’ observations can be treated as approximately independent for the chi-square procedure. This does not establish that commute method and work schedule are independent in the population; that relationship is what the test would investigate under its hypotheses.
Answer: The 10% condition is met because the random sample of 240 residents is only 2% of the population of 12,000. This supports treating the observations as approximately independent. The random condition and expected-count condition still need to be checked separately.
Worked Example: Separate Samples for Homogeneity
Worked Example: Separate Samples for Homogeneity
A fictional research team compares the distribution of three household food-scrap methods across three regions. The team selects a separate random sample without replacement from each region: 120 households from a population of 9,000, 160 households from a population of 14,000, and 200 households from a population of 18,000. Each household reports one usual method.
Identify the quantities. This is a homogeneity setting: separate samples are compared on one categorical response. The 10% condition must be checked for each sample against its own population.
| Region | Sample size \(n\) | Population size \(N\) | 10% of population | Check |
|---|---|---|---|---|
| 1 | 120 | 9,000 | \(0.10(9{,}000)=900\) | \(120 \le 900\) |
| 2 | 160 | 14,000 | \(0.10(14{,}000)=1{,}400\) | \(160 \le 1{,}400\) |
| 3 | 200 | 18,000 | \(0.10(18{,}000)=1{,}800\) | \(200 \le 1{,}800\) |
Check the condition. The sample from Region 1 is no more than 10% of its population; the same is true for Region 2 and Region 3. Equivalently, the population sizes exceed the required minimums: \(9{,}000 \ge 10(120)=1{,}200\), \(14{,}000 \ge 10(160)=1{,}600\), and \(18{,}000 \ge 10(200)=2{,}000\).
Interpret the check. The 10% condition is met for all three samples. The observations in each sample can therefore be treated as approximately independent despite sampling without replacement. This statement does not mean that the three sample sizes must match, and it does not verify the expected counts for the chi-square test.
Answer: The 10% condition is met in each region because each sample is no more than 10% of its own population. The team should still check the random condition and expected counts before proceeding with the test.
Worked Example: A Sample That Is Too Large Relative to Its Population
Worked Example: A Sample That Is Too Large Relative to Its Population
A fictional coastal town has 3,600 households. A researcher randomly selects 450 households without replacement and records each household’s usual method of disposing of yard waste and whether it has a garden. The researcher plans a chi-square test of independence.
Identify the quantities. The sample contains \(n=450\) households, and the population contains \(N=3{,}600\) households.
Check the condition. Ten percent of the population is
The sample size is 450, which is greater than 360, so \(450 \le 360\) is false. The equivalent check also fails: \(10n=10(450)=4{,}500\), but \(N=3{,}600\) is less than 4,500.
Interpret the result. The sample is \(450/3{,}600=0.125\), or 12.5% of the population. It exceeds the 10% limit, so the condition is not met. The sample is not small enough relative to the population to use the 10% check to justify treating observations as approximately independent.
Answer: The 10% condition is not met because the sample of 450 households is 12.5% of the population of 3,600. The researcher should report this limitation rather than claim that the condition is satisfied. The condition’s failure is separate from whether the sample was random and whether expected counts are adequate.
Worked Example: When the 10% Condition Does Not Apply
Worked Example: When the 10% Condition Does Not Apply
A fictional school recruits 100 student volunteers for a study of two lunch-order reminders. Researchers randomly assign 50 volunteers to receive a text reminder and 50 to receive a printed reminder, then record whether each student orders lunch. The researchers plan to compare the response distributions using a chi-square test of homogeneity.
Identify the source of randomness. The description says that students volunteered and were randomly assigned to reminder types. It does not say that they were randomly sampled without replacement from a defined population.
Decide whether to use the 10% check. Because the study description gives random assignment rather than random sampling without replacement, there is no stated sample size \(n\) and finite population size \(N\) to compare for the 10% condition. It would be incorrect to invent a population size just to perform the calculation. In this setting, the 10% condition is not applicable on the information given.
State what still needs checking. The researchers should consider whether the experimental units’ responses can be treated as independent and check the expected counts for the chi-square procedure. Random assignment supports a cause-and-effect conclusion about the reminders for these volunteers if the other conditions are satisfied. Volunteering does not make them a random sample of a broader student population.
Answer: The 10% condition does not apply based on the stated design because it describes random assignment, not sampling without replacement from a finite population. The researchers must still address independence of the experimental units and the other chi-square conditions.
Common Mistakes and AP Exam Tip
- Using the wrong denominator: Compare the sample with the population it was selected from. In a homogeneity test, check each sample against its corresponding population, not against the combined sample size or a different region’s population.
- Using table cells instead of individuals: The sample size is the total number of sampled individuals. It is not the number of cells, categories, or individuals in one cell.
- Confusing two meanings of independence: The 10% condition concerns whether observations can be treated as independent after sampling without replacement. Independence of the two categorical variables is a separate population claim in a test of independence.
- Claiming the condition is met without a population size: If \(N\) is unknown, state that the condition cannot be verified from the information given. Do not assume the population is large enough.
- Treating random assignment as random sampling: Random assignment does not by itself create a sample from a finite population. Apply the 10% condition when sampling without replacement, and address independence of experimental units separately in an experiment.
- Stopping after this check: Meeting the 10% condition does not verify the random condition or the expected-count condition. A complete AP response considers each relevant condition on its own.
Check Your Understanding
For each situation, decide whether the 10% condition is met or applicable, and explain what your decision supports.
- A random sample of 300 households is selected without replacement from a population of 5,000. Check the condition using both equivalent comparisons.
- Researchers select 180 residents without replacement from a town of 1,500. Is the condition met? Show the comparison.
- A homogeneity study uses samples of 90 and 140 people from populations of 4,000 and 1,200, respectively. Check the condition separately for both samples.
- A study randomly assigns volunteers to two treatments but does not randomly sample them from a stated population. Is the 10% condition applicable based on this information? What should still be considered?
- Explain why meeting the 10% condition does not show that two categorical variables are independent.