Two categorical variables and the question of whether they are related. The expected-count rule that decides whether the test is valid at all, why the effect size sits beside the result, and how to report it in APA 7.
A chi-square test of independence asks whether two categorical variables are related. Role level and age group. Department and whether someone left. Region and preferred format.
It compares what you observed against what you would expect if the two variables had nothing to do with each other. If 30% of your whole sample chose a given option, independence predicts about 30% within every group. Chi-square measures how far the actual table sits from that prediction.
A significant chi-square tells you the variables are related. It does not tell you how, or how strongly. The direction lives in the cross-tabulation and the strength lives in the effect size, which is why Quanta puts Cramér's V on the result rather than leaving it to a separate step.
The result opens with four figures: chi-square, df, p, and Cramér's V labelled as the effect. Having the effect size in the headline row rather than buried in a table is the point, and it changes how the result reads.
Consider a real one. Role level by age group on 250 cases returns chi-square = 31.87, df = 20, p = .045, V = 0.18. The p value clears .05, so the conventional reading is a significant association. The V of 0.18 says the association is weak.
Both are true, and reporting only the first would overstate the finding. A p just under threshold paired with a small V is the most common way a categorical result gets oversold.
Note the degrees of freedom. Twenty df means a table with 30 cells. Spread 250 cases across 30 cells and the average expected count is about 8, which means some cells will be well below that. Which brings us to the rule that decides whether any of this is usable.
This is the assumption that decides whether the test is valid, and it is the one most often skipped.
The conventional rule: all expected counts should be at least 5. A commonly accepted relaxation allows up to 20% of cells below 5, provided none falls below 1. The rule concerns expected counts, not observed ones, so you cannot judge it by glancing at the table you collected.
Large tables are where it bites. A 5×6 table on 250 cases is thinly populated by construction, and a p value of .045 from such a table deserves more scrutiny than the same p value from a 2×2 on the same sample.
When the rule fails, chi-square's p is unreliable and usually too small. For a 2×2, use Fisher's exact test. For a larger table, collapse sparse categories into a defensible combined category, and say in the write-up that you did and why.
Independence of observations still applies. Each person contributes to exactly one cell. If the same people are counted twice, use McNemar.
Chi-square scales with sample size. Multiply every cell by ten and chi-square multiplies by roughly ten while the pattern in the data is identical. On a large sample, a trivial association reaches significance.
Cramér's V rescales chi-square to run from 0 to 1 regardless of sample size or table dimensions, which makes it comparable across studies and interpretable on its own.
Read the two together. A V of .18 is a real but weak association. Describing that as evidence of a relationship between role level and age, without the V, invites a reader to imagine something much stronger than the data supports.
Lay the data out one row per person, with a column for each categorical variable. You do not build the cross-tabulation yourself; Quanta constructs it from the raw responses.
Upload categories as text, not as numeric codes. This matters more here than anywhere else. If a response arrives as 1, 2, 3, Quanta reads it as a number rather than a category, and the analysis you want will not be offered.
Open Nonparametric & Categorical and select Chi-square independence from the test list, then configure the two variables in the Setup panel on the right. Results appear in the center.
Read the table dimensions before the p value. The df tells you how many cells the sample is spread across, which is the first clue about whether the expected-count rule is at risk.
Use the action bar to add the result to a report, copy the table, or copy the APA text. Every result carries the ReliCheck Intelligence card for a plain-language explanation computed on your Mac.
Report chi-square with its degrees of freedom and the sample size in parentheses, then p, then the effect size.
Role level and age group were associated, χ²(20, N = 250) = 31.87, p = .045, Cramér's V = .18, indicating a weak association.
The sample size goes inside the parentheses with the degrees of freedom. This is specific to chi-square and one of the most frequent APA formatting errors.
Characterize the effect in words. A significant p with a V of .18 should be described as weak, and saying so protects the finding from being overread.
Describe the pattern and give the percentages that show it. A reader cannot tell from the statistic which group leaned which way.
State that the expected-count rule was met, or state what you did about it. On a table with 30 cells this is not a formality.
Every result carries a ReliCheck Intelligence card with one action: explain this result in plain language. It runs entirely on your Mac with Apple Intelligence. No upload, no API key, nothing to configure. The model only explains the result Quanta has already computed and validated. It never computes a statistic, and it never applies a decision on your behalf.
From the action bar you can add the result to a report, copy the table, or copy the APA text. Exported reports carry the APA table, its note, and a caution specific to that analysis beneath it.
Quanta's engine is pure Swift on Apple's Accelerate framework, with no R or Python at runtime. Every engine is checked against an independent reference rather than internal consistency alone, and the automated suite exceeds 500 checks. Full record on the validation page.
Categorical is part of every Quanta subscription, $9.99 a month or $79.99 a year after the 30-day trial. One price for everyone. See pricing.
The analysis, the report builder, and the on-device explanations all work with the network off. Your dataset is never uploaded.
Chi-square goodness of fit one variable against expected proportions · Fisher exact test when expected counts are small · McNemar for paired categorical data · the complete list on the analyses page.