Troubleshooting
Expected Cell Count Less Than 5 in Chi-Square: What to Do
The warning “cells have expected count less than 5” means the chi-square test's large-sample approximation may be unreliable for your table. For a 2×2 table, switch to Fisher's exact test; for larger tables, combine sparse categories, collect more data, or use an exact or Monte Carlo p-value.
Reviewed by the crosstabs.com methods team · Last updated
What the warning actually means
The Pearson chi-square statistic does not have an exact chi-square distribution. Cell counts are discrete, and the chi-square distribution is continuous — the test relies on a large-sample approximation that becomes accurate as expected counts grow. When expected counts are small, the approximation degrades: the p-value your software reports can be noticeably different from the true probability, usually in the direction of being too small (too eager to declare significance).
That is all the warning says. SPSS prints it as a footnote (“2 cells (50.0%) have expected count less than 5”), R's chisq.test() says “Chi-squared approximation may be incorrect,” and statistics courses teach it as an assumption check. It does not mean your data are wrong or that no test is possible — it means this particular approximation may not be trustworthy and you should reach for a sturdier method.
Note that the rule is about expected counts, not observed ones. A cell can legitimately contain zero observed cases; what matters is the count you would expect under independence.
The conventional rule (Cochran, 1954)
The strictest version of the rule, and the one most textbooks lead with, is that everyexpected count should be at least 5. William Cochran's 1954 review — the source of the convention — is more permissive for tables larger than 2×2:
Rule of thumb
The chi-square approximation is acceptable when no expected count is below 1 and at most 20% of cells have expected counts below 5. Equivalently: at least 80% of cells must have expected counts of 5 or more, and none below 1. For 2×2 tables, where one low cell already exceeds the 20% allowance, all four expected counts should be at least 5.
Cochran himself called the threshold of 5 “arbitrary,” and later simulation work (e.g., Koehler & Larntz) suggests the chi-square test is more robust than the rule implies. But the rule remains the convention reviewers and instructors expect, so when your table violates it, you should either justify keeping the chi-square test or use one of the remedies below.
How an expected count is computed
Expected count under independence
E = (row total × column total) / grand total
- E
- = the count expected in a cell if the two variables were independent
- row total
- = the sum of the cell's row
- column total
- = the sum of the cell's column
- grand total
- = the overall sample size n
Mini example: spotting a low expected count
Suppose 40 people were asked whether they renewed a subscription, split by plan:
Renewed: Basic 18, Premium 6 (row total 24)
Churned: Basic 12, Premium 4 (row total 16)
Column totals: Basic 30, Premium 10 · n = 40
The expected count for the Churned × Premium cell is (16 × 10) / 40 = 4.0. That is below 5, and in a 2×2 table one low cell is 25% of all cells — so the chi-square approximation is suspect here. The observed count in that cell (4) is irrelevant to the check; the expected count is what fails. The right move for this table is Fisher's exact test.
What to do about it
1. Use Fisher's exact test (2×2 tables)
For a 2×2 table this is the standard fix. Fisher's exact test computes the p-value directly from the hypergeometric distribution, with no large-sample approximation, so low expected counts are not a problem. Run one in your browser with our Fisher's exact test calculator, or see chi-square vs Fisher's exact for when each applies.
2. Combine sparse categories
If a row or column is thinly populated — say, an “Other” response chosen by three people — merge it with a substantively similar category and recompute. This raises expected counts throughout the affected row or column. Only merge categories that make sense together: combining “Strongly disagree” with “Disagree” is defensible; combining it with “Neutral” usually is not. Decide on merges before looking at which combination produces a significant result.
3. Collect more data
Expected counts scale with sample size, so a larger sample is the cleanest cure when it is feasible. If a cell's expected count is 2.5 at n = 80, doubling the sample (with similar marginals) pushes it to about 5. This is worth stating in a limitations section even when you cannot act on it.
4. Use an exact or Monte Carlo p-value (larger tables)
For r×c tables bigger than 2×2, most packages offer exact tests (the Freeman–Halton extension of Fisher's test) or Monte Carlo simulation of the chi-square p-value — in R, chisq.test(x, simulate.p.value = TRUE); in SPSS, the Exact Tests module. These keep your original categories intact while removing the reliance on the large-sample approximation.
What not to do
Do not simply ignore the warning and report the chi-square p-value as if nothing happened — especially when the p-value sits near your significance threshold, which is exactly where approximation error matters most. And do not drop the offending rows of data; deleting real observations to silence a warning biases the table.
Yates' continuity correction deserves a mention: it is sometimes presented as the fix for small 2×2 tables, but it tends to overcorrect, making the test conservative. With Fisher's exact test a few clicks away, there is little reason to prefer the corrected chi-square. Our chi-square calculator reports expected counts for every cell so you can check the rule directly instead of guessing.
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Open the workspace →Frequently asked questions
- Can I still run a chi-square test if one cell has an expected count less than 5?
- Often yes. The widely used Cochran criterion allows the chi-square test as long as no expected count is below 1 and no more than 20% of cells have expected counts below 5. A single cell slightly under 5 in a larger table rarely invalidates the test. For a 2×2 table, where one low cell is already 25% of the table, switch to Fisher's exact test instead.
- Does the 'expected count less than 5' rule apply to observed or expected counts?
- Expected counts only. An observed count of 0 or 2 is fine by itself — the rule concerns the expected frequency E = (row total × column total) / grand total, which is what the chi-square approximation depends on. SPSS's warning footnote reports expected counts, not observed ones.
- Is Fisher's exact test always better than chi-square for small samples?
- For 2×2 tables with low expected counts, Fisher's exact test is the standard replacement because it computes an exact p-value with no large-sample approximation. For larger tables, exact or Monte Carlo versions of the chi-square test serve the same purpose; Fisher's test generalizes to r×c tables (the Freeman–Halton extension) but is computationally heavier and not available in every package.
References & further reading
- Cochran, W. G. (1954). Some methods for strengthening the common χ² tests. Biometrics, 10(4), 417–451.
- Agresti, A. Categorical Data Analysis — small-sample inference for contingency tables.
- Fisher's exact test — Wikipedia