Effect size measure
Phi Coefficient Calculator
The phi coefficient (φ)measures the strength of association between two binary variables in a 2×2 contingency table. It ranges from −1 to +1 (or 0 to 1 in magnitude), where 0 means no association and ±1 means a perfect one — and for a 2×2 table it is mathematically identical to Cramér's V.
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Open the workspace →What is the phi coefficient?
Phi is the categorical analogue of a correlation coefficient for two yes/no variables. Coded as 0 and 1, the phi coefficient is exactly the Pearson correlation between those two binary columns, which is why it carries a sign: positive φ means the two “1” categories occur together more than chance, negative φ means they avoid each other.
Because it is built from the chi-square statistic, phi pairs naturally with a chi-square test: the test tells you whether two variables are associated, and phi tells you how strongly.
Formula
Definition
φ = (ad − bc) / √((a+b)(c+d)(a+c)(b+d))
- a, b, c, d
- = the four cell counts of a 2×2 table [[a, b], [c, d]]
- equivalently
- = |φ| = √(χ² / n), where n is the grand total
Worked example
Worked example
Suppose 110 people are cross-tabulated by whether they saw an ad (rows) and whether they bought (columns):
[[30, 10], [10, 60]] → a=30, b=10, c=10, d=60
Numerator: ad − bc = (30×60) − (10×10) = 1800 − 100 = 1700.
Denominator: √((40)(70)(40)(70)) = √7,840,000 = 2800.
φ = 1700 / 2800 = 0.61.
A phi of 0.61 is a strong positive association — seeing the ad and buying go together far more than chance.
When to use it
Use it when
- Both variables are binary (exactly two categories each).
- You have a 2×2 contingency table and want an effect size to accompany a chi-square test.
- You want a signed measure that shows direction, not just strength.
Not the right tool when
- The table is larger than 2×2 — use Cramér's V instead.
- Either variable is continuous — use the Pearson correlation.
- You only care about an exact p-value for a tiny sample — use Fisher's exact test for that, then report phi.
How to interpret it
Rule of thumb
As a rough guide (Cohen), |φ| ≈ 0.1 is a small effect, ≈ 0.3 is medium, and ≈ 0.5 or above is large. The sign indicates direction; the magnitude indicates strength.
Frequently asked questions
- Is the phi coefficient the same as Cramér's V?
- For a 2×2 table, yes — phi and Cramér's V are mathematically identical. For tables larger than 2×2 they differ, and Cramér's V is the right choice because it stays bounded between 0 and 1.
- Can the phi coefficient be negative?
- Yes. Computed as (ad − bc) over the square-root term, phi ranges from −1 to +1 and carries a sign that shows the direction of association. The magnitude √(χ²/n) is always between 0 and 1.
- What is a good phi coefficient value?
- There is no universal threshold, but Cohen's benchmarks are common: about 0.1 is small, 0.3 is medium, and 0.5 or more is large. Always interpret it alongside the chi-square p-value and your context.
- How is phi related to the Pearson correlation?
- If you code each binary variable as 0 and 1, the phi coefficient is exactly the Pearson correlation coefficient between the two columns.
References & further reading
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.).
- Phi coefficient — Wikipedia
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