Enter the four cells of your 2×2 contingency table — Cell A (both yes), Cell B (var1 yes / var2 no), Cell C (var1 no / var2 yes), and Cell D (both no) — and the Phi Coefficient Calculator returns the phi (φ) value, chi-square statistic, total N, and a plain-language interpretation of the association strength between your two binary variables.
Phi Coefficient (φ)
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Chi-Square (χ²)
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Total Sample Size (N)
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Association Strength
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The Phi Coefficient (φ) is a measure of association between two binary (dichotomous) variables arranged in a 2×2 contingency table. It ranges from -1 to +1, where -1 indicates a perfect negative association, +1 indicates a perfect positive association, and 0 indicates no association. It is mathematically equivalent to Pearson's correlation coefficient when both variables are binary.
The Phi Coefficient is calculated as φ = (AD − BC) / √[(A+B)(C+D)(A+C)(B+D)], where A, B, C, D are the four cell counts of the 2×2 table. It can also be computed as φ = √(χ²/N), where χ² is the chi-square statistic and N is the total sample size.
A φ value near 0 suggests no association, values around ±0.1 are considered small, values around ±0.3 are moderate, and values of ±0.5 or above are considered large or strong associations. The sign (positive or negative) indicates the direction of the association between the two variables.
Use the Phi Coefficient specifically when both variables are dichotomous (binary) and your data fits a 2×2 contingency table. For larger contingency tables, Cramér's V is more appropriate. For continuous variables, use Pearson's r. For ordinal data, consider Spearman's rho or Kendall's tau.
The Phi Coefficient requires two dichotomous (binary) variables, data organized in a 2×2 contingency table, and independent observations. Each participant or case should appear in only one cell of the table, and the total sample size should generally be at least 20 for meaningful results.
The null hypothesis (H₀) states that φ = 0, meaning there is no association between the two dichotomous variables. A significant result (typically assessed via the associated chi-square test) suggests the observed association is unlikely to be due to chance alone.
The Phi Coefficient can only reach its extreme values of -1 or +1 when the marginal totals of the table are equal. When marginals are unequal, the maximum possible absolute value of φ may be less than 1, which can make comparisons across studies with different marginal distributions difficult.
The Phi Coefficient is directly related to the chi-square statistic by the formula φ = √(χ²/N). This means you can derive φ from any chi-square test on a 2×2 table, and squaring φ (φ²) gives you the proportion of variance shared between the two binary variables.