Example Data Table
| Treatment Result |
Improved |
No Change |
Worse |
| Treatment A |
8 |
2 |
1 |
| Treatment B |
3 |
6 |
2 |
| Treatment C |
1 |
4 |
7 |
Formula Used
The calculator uses the Fisher Freeman Halton extension for a 3x3 table. It conditions on fixed row and column totals.
Table probability: P = [product of row total factorials × product of column total factorials] ÷ [grand total factorial × product of cell factorials].
Two-sided exact p-value: Sum probabilities for all fixed-margin tables with probability less than or equal to the observed table probability.
Expected count: Eij = row total i × column total j ÷ grand total.
Cramer’s V: V = square root of chi square ÷ [grand total × 2].
How to Use This Calculator
Enter raw counts only. Do not enter percentages or rates.
Add optional row and column labels. They make the report easier to read.
Select a p-value method. The usual option is two-sided probability ordering.
Set alpha, such as 0.05. Then press Calculate.
Review the result above the form. Use CSV or PDF to save the report.
Understanding the 3x3 Fisher Exact Test
A 3x3 Fisher exact test studies association between two categorical variables. Each variable has three groups. The method is useful when counts are small. It does not rely on a large sample approximation. Instead, it conditions on the row totals and column totals. Then it compares the observed table with every possible table sharing those margins.
Why Exact Testing Helps
A chi square test can be fast. Yet it may be unreliable when expected counts are low. Fisher testing avoids that weakness. It calculates table probabilities from the hypergeometric distribution. For a 3x3 table, this is often called the Fisher Freeman Halton extension. The result is an exact conditional p value. It answers a clear question. How unusual is this table, if row and column categories are independent?
What This Calculator Reports
This calculator accepts nine nonnegative cell counts. It checks the grand total. It computes margins, expected counts, table probability, and the exact p value. It also reports a chi square statistic and Cramer’s V. These extra values help describe strength and direction of evidence. They should not replace the exact p value. They add context for reporting.
Interpreting the Output
A small p value suggests the observed pattern is unlikely under independence. Use your selected alpha level as a decision guide. For example, p below 0.05 often indicates statistical evidence of association. A larger p value does not prove independence. It means the table is not unusual enough under the fixed margin model.
Good Practice
Use meaningful row and column labels. Avoid mixing rates and counts. Enter raw frequencies only. Review expected counts before reporting. If totals are large, exact enumeration can take more time. Keep practical sample sizes when using a web form. Include the p value, margins, test type, and any selected option in your report.
Limits and Notes
The two sided rule used here sums all tables with probabilities less than or equal to the observed probability. This is a common exact approach. Other software may use different ordering rules. Results can differ slightly. State the rule used. This keeps your statistical report transparent, repeatable, and easier to review. Always verify data entry before making a formal decision.
FAQs
What is a 3x3 Fisher exact test?
It is an exact test for a contingency table with three row groups and three column groups. It checks whether the categories appear associated while holding margins fixed.
When should I use this instead of chi square?
Use it when sample sizes are small or expected counts are low. It is also useful when you need an exact conditional p value.
Can I enter percentages?
No. Enter raw frequency counts only. Percentages, rates, averages, and rounded proportions are not valid inputs for this exact test.
What does the exact p-value mean?
It shows how unusual the observed table is under independence, given the same row and column totals. Smaller values suggest stronger evidence of association.
What is mid-p?
Mid-p subtracts half of the observed probability mass from the usual exact tail. It can be less conservative, but should be reported clearly.
Why is there an enumeration limit?
A 3x3 exact test may require checking many possible tables. The limit helps prevent very slow page loads for large totals.
What is Cramer’s V?
Cramer’s V is an effect size for categorical association. It gives extra context, but it does not replace the exact p-value.
Can software results differ?
Yes. Two-sided exact tests can use different table ordering rules. Always state the method when reporting your results.