Chi-Square Assumptions and Expected Counts
Chi square assumptions expected counts determine whether a chi-square reference distribution is trustworthy. This page is not another calculator bank: it is a condition-checking and expected-count diagnostic guide for categorical inference.
Quick answer: the conditions to check
A defensible contingency-table chi-square test begins with categorical count data, observations that are independent enough for the sampling design, and expected counts large enough for the chi-square approximation. For a random sample taken without replacement from a finite population, the familiar 10% condition is a practical way to support independence between sampled observations.
For homogeneity, think about independent random samples or randomized groups whose categorical distributions are compared. For independence, think about one random sample in which two categorical variables are measured on each observational unit. The same expected-count formula may appear in both settings, but the study design determines which population statement the conclusion can support.
Chi square assumptions expected counts are not a ceremonial checklist. A violated condition changes what the p-value means. If expected counts are too small, if observations are clustered or repeated, or if the sample is not connected to the population named in the conclusion, a precise calculator output can still be statistically misleading.
How expected counts encode the null model
In a contingency table, the expected count for a cell is row total × column total ÷ grand total. This formula creates the table we would anticipate if the row and column classifications were unrelated, or if compared populations had the same categorical distribution. Expected counts therefore come from the null model and the margins, not from replacing observations with averages.
An expected count can be non-integer even though an observed count must be an integer. For example, an expected value of 17.6 means that under repeated comparable samples the long-run null-model average for that cell would be 17.6; it is not claiming that 0.6 of a person was observed. Treating expected counts as model quantities prevents an unnecessary rounding error before the statistic is computed.
Every expected count should be checked in its actual cell. A table may have a large total sample size but still produce a small expected value in a rare category. The relevant adequacy question is cell-specific, not simply whether n exceeds some overall threshold.
| Question | Diagnostic |
|---|---|
| Are the entries counts? | Observed cells should be frequencies for mutually exclusive category combinations. |
| Do expectations preserve margins? | Expected row/column sums must reproduce the observed margins. |
| Are observations independent enough? | Inspect sampling, clustering, repeated measures, and finite-population fraction. |
| Are expected counts sufficiently large? | Use the course/test condition rather than relying on total n alone. |
| Does the conclusion match the design? | Random sampling supports population generalization; random assignment supports causal comparison. |
Condition map for homogeneity and independence
For a chi-square test of homogeneity, the data typically come from two or more independently selected groups, and the goal is to compare the distribution of one categorical response across those populations or treatments. Independence between groups matters: the same person should not appear in multiple independently analyzed groups unless the method explicitly accounts for pairing.
For a chi-square test of independence, a single sample supplies two categorical measurements per unit. The null hypothesis says the two variables are independent in the population represented by the sample. Here, “independence” in the hypothesis is conceptually different from “independent observations” as a condition. The former is what is tested between variables; the latter concerns the data-generating process.
Random assignment can justify a causal statement about treatment effects even when the subjects were not randomly sampled from a broad population. Random sampling can justify population generalization but does not, by itself, turn an observational association into causation. Keeping those scopes separate is part of checking assumptions, not an optional writing flourish.
Twenty expected-count and design diagnostic cases
Diagnostic case 1: Repeated cafeteria visits
Setup. Each of 45 students reports lunch choice on four different days, and all 180 rows are treated as independent students.
Assessment. Repeated measurements on the same student create dependence; a simple chi-square independence test on 180 independent units is not justified.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to repeated cafeteria visits.
Diagnostic case 2: Rare response category
Setup. A 3×4 table has n=420, but one expected cell count is 2.8.
Assessment. Large total n does not repair a cell with a very small expected count; inspect or combine substantively defensible categories or use a more appropriate method.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to rare response category.
Diagnostic case 3: Simple random sample
Setup. A district takes an SRS of 180 students from 8,000 and records grade band and transport mode.
Assessment. The sampling fraction is well below 10%, supporting approximate independence between sampled students; then inspect all expected counts.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to simple random sample.
Diagnostic case 4: Cluster sample ignored
Setup. Ten classrooms are sampled and every child in those classrooms is analyzed as if all children were independently sampled.
Assessment. Students within a classroom may be correlated, so the simple independence condition is questionable even when expected counts are large.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to cluster sample ignored.
Diagnostic case 5: Randomized treatment groups
Setup. Participants are randomly assigned to three reminder systems, then response category is recorded once.
Assessment. Random assignment supports causal treatment comparison if implementation is sound; the homogeneity-style table must still have adequate expected counts.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to randomized treatment groups.
Diagnostic case 6: Convenience website poll
Setup. Visitors who choose to answer an online poll are compared by device and opinion.
Assessment. Expected counts may be excellent, but voluntary response limits population generalization because the design does not create a representative random sample.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to convenience website poll.
Diagnostic case 7: Expected-count rounding
Setup. A student rounds each expected count to the nearest integer before calculating χ².
Assessment. Expected counts should generally remain unrounded during computation; early rounding can distort cell contributions and the final statistic.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to expected-count rounding.
Diagnostic case 8: Percent table input
Setup. Only row percentages are available and the original group sample sizes are unknown.
Assessment. Percentages alone do not recover the frequency scale needed for the standard chi-square statistic unless the corresponding counts/sample sizes are known.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to percent table input.
Diagnostic case 9: Duplicated records
Setup. A data export accidentally includes 14 participants twice.
Assessment. Duplicated units violate the intended one-observation-per-unit structure and inflate apparent information; clean the data before inference.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to duplicated records.
Diagnostic case 10: Matched siblings
Setup. Pairs of siblings are deliberately recruited, then analyzed as 120 independent individuals.
Assessment. Within-pair similarity can violate independent-observation assumptions; a method recognizing pairing or clustering may be needed.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to matched siblings.
Diagnostic case 11: Very large balanced table
Setup. A 2×3 table from independent samples has all expected counts above 40.
Assessment. The expected-count approximation is comfortably supported; proceed to the statistic after checking design and scope.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to very large balanced table.
Diagnostic case 12: Structural zero
Setup. A category combination is impossible by definition, creating an expected-count formula that does not reflect the design.
Assessment. A structural zero is not the same as a randomly small cell and may require a model/table definition that respects the impossibility.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to structural zero.
Diagnostic case 13: Post-hoc category merging
Setup. Categories are merged only after seeing which cells produce the largest χ² contributions.
Assessment. Data-driven merging can alter the hypothesis after inspecting evidence; category definitions should be substantively defensible and preferably specified before testing.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to post-hoc category merging.
Diagnostic case 14: Different sampling frames
Setup. Two population samples were drawn using different frames with markedly different coverage.
Assessment. A homogeneity comparison can be distorted by frame differences; design comparability matters in addition to arithmetic assumptions.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to different sampling frames.
Diagnostic case 15: Ten-percent check
Setup. An SRS of 120 is drawn without replacement from a population of 900.
Assessment. Because 120 exceeds 10% of 900, the usual simple finite-population independence approximation is not automatically supported.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to ten-percent check.
Diagnostic case 16: One response per person
Setup. A national SRS records region and preferred news format once for each respondent.
Assessment. One row per independently sampled person is appropriate; next verify expected counts and whether the sampling frame supports the population conclusion.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to one response per person.
Diagnostic case 17: Missing category values
Setup. Respondents with missing outcomes are silently dropped and missingness is related to group.
Assessment. The analyzed table may no longer represent the target populations; missing-data mechanisms can create bias even if chi-square mechanics are valid.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to missing category values.
Diagnostic case 18: Sparse 5×6 table
Setup. A table has 30 cells but only 75 observations.
Assessment. Many small expected values are likely; inspect every cell rather than assuming the approximation because the table has many categories.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to sparse 5×6 table.
Diagnostic case 19: Survey weights ignored
Setup. A complex probability survey supplies sampling weights, but raw unweighted counts are used in a simple chi-square test.
Assessment. Complex survey design can require design-based methods; the ordinary test may not reflect unequal selection probabilities or clustering.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to survey weights ignored.
Diagnostic case 20: Random sample, observational exposure
Setup. A random sample compares smoking category with respiratory-symptom category.
Assessment. Random sampling can support association inference to the sampled population, but without random assignment the association does not by itself establish a causal effect of smoking category.
What to write. Identify the exact condition involved, tie it to the study design rather than reciting a slogan, and explain how the issue affects the validity or scope of inference. This produces a stronger response than simply listing “random, 10%, expected counts” without connecting those words to random sample, observational exposure.
What to do when expected counts are small
When an expected count is too small for the intended chi-square approximation, the first task is diagnosis, not automatic deletion. Ask whether the rare category is scientifically meaningful, whether categories were defined too finely, whether a larger sample is feasible, and whether an exact or alternative modeling approach is more appropriate for the design.
Combining categories can be defensible when the merged category has a coherent substantive meaning established independently of the observed test result. Combining “just enough” categories to force a desired p-value is not defensible. The category system defines the hypothesis, so changing categories changes the question being tested.
Never replace a small expected count with 5. Expected counts are determined by the null model and margins; manually inflating a cell destroys the reference model. Likewise, dropping a cell changes totals and can change both the null hypothesis and df.
Assumptions determine the scope of the conclusion
A well-calculated statistic cannot compensate for a mismatched target population. If data came from a convenience sample at one school, the conclusion should not silently expand to all U.S. students. If subjects were randomly assigned to treatments but recruited as volunteers, causal inference about those participants can be stronger than population generalization.
For observational contingency tables, association is the appropriate language. Statements such as “the variables are associated,” “the distributions differ,” or “there is evidence of a relationship” respect the design. Causal verbs such as causes, leads to, or improves require a design that supports causal attribution.
Condition checking belongs before the p-value interpretation. If a serious violation is discovered after software produces a tiny p-value, the correct response is not to ignore the violation because the evidence looks strong. The reference distribution itself may be unreliable under the violated assumptions.
Chi-square assumptions multiple-choice practice
Question 1. Conditions and expected counts
A table has expected counts 7.2, 9.4, 11.8, and 13.6. Which concern is least relevant?
Answer: D
Formatting does not affect the statistical reference model; the other issues can.
Question 2. Conditions and expected counts
What is the expected count formula for a contingency-table cell?
Answer: A
The null expectation preserves both margins through the product-over-grand-total formula.
Question 3. Conditions and expected counts
Which is a condition issue rather than the null hypothesis itself?
Answer: B
Independent observations concern data collection; variable independence or equal distributions are hypotheses.
Question 4. Conditions and expected counts
An SRS of 80 is drawn from 500 people without replacement. Which check raises concern?
Answer: B
80 is 16% of 500, so the usual 10% finite-population condition is not met.
Question 5. Conditions and expected counts
Why should expected counts generally not be rounded before calculating χ²?
Answer: B
Expected counts are model quantities and can be decimal; retaining precision improves the calculation.
Question 6. Conditions and expected counts
A very large n guarantees which of the following?
Answer: C
Sparse categories or biased designs can persist even with a large total sample.
Question 7. Conditions and expected counts
Which situation most directly threatens independent observations?
Answer: B
Repeated rows from the same unit are correlated rather than independent observational units.
Question 8. Conditions and expected counts
A randomized experiment compares treatment with outcome category. What can random assignment support?
Answer: B
Random assignment targets causal treatment comparison, not automatic population representation.
Question 9. Conditions and expected counts
Expected counts sum to 198 while observed counts sum to 200. What should you suspect first?
Answer: A
Expected counts for the full table should reallocate the same grand total.
Question 10. Conditions and expected counts
Why is a structural zero different from a small random count?
Answer: A
Structural impossibility should be represented in the model rather than treated as ordinary sampling sparsity.
Question 11. Conditions and expected counts
A voluntary-response poll has excellent expected counts. What remains a major concern?
Answer: A
Good approximation conditions do not cure a biased recruitment mechanism.
Question 12. Conditions and expected counts
When can category merging be most defensible?
Answer: B
The merge should represent a meaningful revised question, not a post-hoc significance strategy.
Question 13. Conditions and expected counts
What does the 10% condition mainly support in an SRS without replacement?
Answer: A
Sampling a small fraction makes dependence from sampling without replacement negligible.
Question 14. Conditions and expected counts
A 2×3 table has all expected counts above 30 but data came from paired observations. Which statement is best?
Answer: B
Different assumptions address different features; one satisfied condition cannot repair another.
Question 15. Conditions and expected counts
Which conclusion is safest for a random observational sample?
Answer: B
Random sampling can support population association inference but not causal attribution by itself.
Question 16. Conditions and expected counts
A cell expectation is 2.1. What is the best first response?
Answer: C
Small expected counts are a methodological warning requiring a principled response.
Question 17. Conditions and expected counts
Which pair should match when expected counts are calculated correctly?
Answer: A
Expected values preserve the sample’s total count even though individual cells differ.
Question 18. Conditions and expected counts
Why write conditions in context?
Answer: A
Conditions are claims about the data-generating process, so contextual justification is essential.
Chi-square assumptions free-response practice
FRQ 1. Repeated observations
A health study records symptom category weekly for each participant and constructs one large contingency table as if each week came from a different person. Evaluate the independence condition.
Model response
The observational units in the analysis are weekly records, but records from the same participant are likely correlated. Treating them as independent can understate uncertainty and invalidate the ordinary chi-square reference distribution.
A better design-specific analysis should account for repeated measurements or redefine the observational unit so each person contributes appropriately.
FRQ 2. Sparse category
A 2×5 table has expected counts 18.4, 12.7, 8.1, 4.3, 6.5 in one row. Explain the condition concern and one principled response.
Model response
The expected count 4.3 signals that the standard large-sample approximation may be questionable under the course rule being applied. Total sample size alone does not remove the problem.
Possible responses include collecting more data, using an appropriate exact/alternative procedure, or combining categories only when a substantive preexisting rationale supports the merge.
FRQ 3. Random sample scope
A county takes an SRS of 200 households from 12,000 and records heating type and housing type. Explain what the design supports.
Model response
The sample is a small fraction of the population, so the 10% check supports approximate independence from sampling without replacement, assuming the SRS was implemented correctly.
Because this is observational, a significant association can be generalized to the county population represented by the frame, but it does not establish that one housing characteristic causes the other.
FRQ 4. Expected-count verification
A 2×3 table has row totals 80 and 120, column totals 50, 70, 80. Compute the first row expected counts.
Model response
With grand total 200, the first-row expectations are 80×50/200=20, 80×70/200=28, and 80×80/200=32.
They sum to 80, matching the first-row total, which is an important arithmetic check before calculating χ².
FRQ 5. Random assignment
Volunteers are randomly assigned to three notification systems and response category is measured once. Distinguish causal scope from population scope.
Model response
Random assignment supports a causal comparison among the assigned notification systems for the study participants, provided treatment implementation and measurement are sound.
Because the participants were volunteers rather than a probability sample, broad population generalization requires caution even if expected counts and independence conditions are otherwise adequate.
FRQ 6. Condition failure after output
Software produces p=.001, but an audit reveals that 30% of the rows are duplicate participant records. Explain why the small p-value should not be reported uncritically.
Model response
Duplicated rows violate the intended independence/information structure and artificially weight some participants, so the reference distribution and effective sample information are not what the software assumed.
The data should be corrected and the analysis rerun before interpreting the p-value. A tiny numerical result does not override a serious data-generating violation.
Extended topic-specific mastery cases
Sampling Independence mastery extension 1
Mastery diagnostic 1 examines sampling independence. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes sampling independence a reasoned diagnostic rather than a memorized checklist.
Finite-Population Check mastery extension 2
Mastery diagnostic 2 examines finite-population check. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes finite-population check a reasoned diagnostic rather than a memorized checklist.
Sparse Expected Cell mastery extension 3
Mastery diagnostic 3 examines sparse expected cell. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes sparse expected cell a reasoned diagnostic rather than a memorized checklist.
Cluster Dependence mastery extension 4
Mastery diagnostic 4 examines cluster dependence. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes cluster dependence a reasoned diagnostic rather than a memorized checklist.
Category Definition mastery extension 5
Mastery diagnostic 5 examines category definition. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes category definition a reasoned diagnostic rather than a memorized checklist.
Scope Of Inference mastery extension 6
Mastery diagnostic 6 examines scope of inference. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes scope of inference a reasoned diagnostic rather than a memorized checklist.
Sampling Independence mastery extension 7
Mastery diagnostic 7 examines sampling independence. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes sampling independence a reasoned diagnostic rather than a memorized checklist.
Finite-Population Check mastery extension 8
Mastery diagnostic 8 examines finite-population check. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes finite-population check a reasoned diagnostic rather than a memorized checklist.
Sparse Expected Cell mastery extension 9
Mastery diagnostic 9 examines sparse expected cell. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes sparse expected cell a reasoned diagnostic rather than a memorized checklist.
Cluster Dependence mastery extension 10
Mastery diagnostic 10 examines cluster dependence. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes cluster dependence a reasoned diagnostic rather than a memorized checklist.
Category Definition mastery extension 11
Mastery diagnostic 11 examines category definition. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes category definition a reasoned diagnostic rather than a memorized checklist.
Scope Of Inference mastery extension 12
Mastery diagnostic 12 examines scope of inference. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes scope of inference a reasoned diagnostic rather than a memorized checklist.
Sampling Independence mastery extension 13
Mastery diagnostic 13 examines sampling independence. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes sampling independence a reasoned diagnostic rather than a memorized checklist.
Finite-Population Check mastery extension 14
Mastery diagnostic 14 examines finite-population check. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes finite-population check a reasoned diagnostic rather than a memorized checklist.
Sparse Expected Cell mastery extension 15
Mastery diagnostic 15 examines sparse expected cell. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes sparse expected cell a reasoned diagnostic rather than a memorized checklist.
Cluster Dependence mastery extension 16
Mastery diagnostic 16 examines cluster dependence. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes cluster dependence a reasoned diagnostic rather than a memorized checklist.
Category Definition mastery extension 17
Mastery diagnostic 17 examines category definition. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes category definition a reasoned diagnostic rather than a memorized checklist.
Scope Of Inference mastery extension 18
Mastery diagnostic 18 examines scope of inference. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes scope of inference a reasoned diagnostic rather than a memorized checklist.
Sampling Independence mastery extension 19
Mastery diagnostic 19 examines sampling independence. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes sampling independence a reasoned diagnostic rather than a memorized checklist.
Finite-Population Check mastery extension 20
Mastery diagnostic 20 examines finite-population check. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes finite-population check a reasoned diagnostic rather than a memorized checklist.
Sparse Expected Cell mastery extension 21
Mastery diagnostic 21 examines sparse expected cell. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes sparse expected cell a reasoned diagnostic rather than a memorized checklist.
Cluster Dependence mastery extension 22
Mastery diagnostic 22 examines cluster dependence. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes cluster dependence a reasoned diagnostic rather than a memorized checklist.
Category Definition mastery extension 23
Mastery diagnostic 23 examines category definition. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes category definition a reasoned diagnostic rather than a memorized checklist.
Scope Of Inference mastery extension 24
Mastery diagnostic 24 examines scope of inference. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes scope of inference a reasoned diagnostic rather than a memorized checklist.
Sampling Independence mastery extension 25
Mastery diagnostic 25 examines sampling independence. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes sampling independence a reasoned diagnostic rather than a memorized checklist.
Finite-Population Check mastery extension 26
Mastery diagnostic 26 examines finite-population check. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes finite-population check a reasoned diagnostic rather than a memorized checklist.
Sparse Expected Cell mastery extension 27
Mastery diagnostic 27 examines sparse expected cell. Imagine receiving only the study description and a contingency table, with no p-value yet. Write the observational unit, sampling or assignment mechanism, target population, and the two categorical variables. Then calculate or inspect the expected counts without rounding and identify the smallest expectation. Explain separately whether the sampling design supports independent observations, whether a finite-population check is relevant, whether clustering or repeated measures exists, and whether the expected-count approximation is plausible. If a condition is questionable, propose a principled design or modeling response rather than changing the data to force a test. Conclude by stating what kind of claim—association, population difference, or causal treatment effect—the design could support if the remaining conditions are acceptable. Keeping these layers separate makes sparse expected cell a reasoned diagnostic rather than a memorized checklist.
Chi-square assumptions FAQs
Are expected counts observed data? No. They are null-model quantities computed from margins or specified probabilities.
Does a large sample guarantee good expected counts? No. Rare categories can stay sparse even when total n is large.
Is the 10% condition the same as the expected-count condition? No. The 10% condition addresses dependence from sampling without replacement; expected-count conditions address the chi-square approximation.
Can random sampling establish causation? Not by itself. Random sampling supports generalization; random assignment is the key design mechanism for causal treatment comparisons.
Should categories be combined just to make the test significant? No. Category definitions should be substantively defensible rather than selected after looking at the p-value.
Continue with related AP Statistics resources
Chi Square Assumptions Expected Counts review focus
The phrase chi square assumptions expected counts names this page’s specific purpose. Use chi square assumptions expected counts as the focus when deciding which workflow, conditions, interpretation, or legacy-status guidance belongs here rather than on a neighboring AP Statistics page.
For final review, return to the opening explanation and verify that you can explain chi square assumptions expected counts in context without relying on a memorized label alone.
