Answer First
The Chi‑Square Test is an approximation that works well for large samples, while Fisher’s Exact Test computes the exact probability of the observed table under the null hypothesis. Fisher’s Exact Test is preferred when sample sizes are small or expected counts are below 5.
Problem Setup
Both tests analyze a 2×2 contingency table:
\[ \begin{array}{c|cc|c} & B_1 & B_2 & \text{Row Totals} \\ \hline A_1 & a & b & a+b \\ A_2 & c & d & c+d \\ \hline \text{Column Totals} & a+c & b+d & n \end{array} \]
Chi‑Square Test
Uses the statistic:
\[ \chi^2 = \sum \frac{(O – E)^2}{E} \]
Expected counts:
\[ E_{ij} = \frac{(\text{row total})(\text{column total})}{n} \]
Fisher’s Exact Test
Computes the exact probability of the observed table:
\[ P = \frac{(a+b)!(c+d)!(a+c)!(b+d)!}{a!b!c!d!n!} \]
and sums probabilities of all tables as extreme or more extreme.
Step-by-Step Solution
1. Fisher’s Exact Test is exact
It calculates the exact probability of the observed table under the null hypothesis of independence.
2. Chi‑Square is an approximation
It uses a chi‑square distribution to approximate the sampling distribution of the test statistic.
3. Fisher’s is preferred for small samples
Use Fisher’s when any expected count is below 5.
4. Chi‑Square is preferred for large samples
When all expected counts are ≥ 5, the chi‑square approximation is accurate and faster.
5. Fisher’s is conservative for large samples
It tends to produce slightly larger p‑values when n is large.
Intuition
The Chi‑Square Test says: “If the sample is big enough, the chi‑square curve is a good approximation.” Fisher’s Exact Test says: “Let’s compute the exact probability instead of approximating.”
Common Exam Mistakes
- Using Chi‑Square when expected counts are too small.
- Thinking Fisher’s is only for tiny samples (it works for any size).
- Confusing observed counts with expected counts.
- Assuming Fisher’s and Chi‑Square always give the same p‑value.
Why This Matters
Fisher’s Exact Test is essential in medical studies, A/B testing, HR analytics, and any situation with small sample sizes. The Chi‑Square Test is widely used for large surveys, marketing data, and operational analytics. Choosing the correct test ensures valid conclusions about independence.
Final Summary
The Chi‑Square Test is a large‑sample approximation, while Fisher’s Exact Test computes the exact probability of the observed table. Use Fisher’s when expected counts are small and Chi‑Square when sample sizes are large.
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