StatReason

Phi Coefficient Calculator

Summarize how strongly two binary variables are associated in a 2 by 2 table, on a signed scale similar to a correlation coefficient, directly derived from the table's chi-square statistic.

What this answers

This calculator answers "how strongly, and in which direction, are these two binary variables associated?" for a table with exactly 2 rows and 2 columns, giving a single signed number similar in spirit to a Pearson correlation coefficient.

How it is calculated

Phi is the square root of the table's chi-square statistic divided by the total count, with its sign taken from whether the diagonal cells (a and d) outweigh the off-diagonal cells (b and c) or vice versa. For a 2 by 2 table specifically, phi is mathematically equivalent to the Pearson correlation you would get from coding both variables 0/1 and correlating them directly, though phi is computed here straight from the chi-square relationship rather than by that route.

Worked example

For a table with 10 and 0 in the first row and 0 and 10 in the second row, a perfectly clean diagonal pattern, phi comes out to exactly 1, indicating a perfect positive association: knowing the row perfectly predicts the column. Reversing the diagonal pattern would give phi equal to exactly negative 1 instead.

Assumption audit

Calculated from your data: the table's chi-square statistic and the resulting signed phi coefficient, bounded between negative 1 and 1.
Evidence to review: your table's orientation; phi's sign depends entirely on which categories you assigned to which row and column, so a positive or negative sign only means something once you know what each row and column represents.
You must verify: that both variables are genuinely binary categories, not artificially dichotomized continuous variables, and that observations are independent of one another.

What this result does not mean

Phi measures association strength and direction, not causation, and it is only valid for a 2 by 2 table specifically; a larger contingency table needs the Cramer's V Calculator instead, which generalizes the same underlying idea to any table size but drops the signed direction, since direction is not a well-defined concept beyond 2 categories per variable.

Limitations

Like any measure derived from chi-square, phi can be affected by very small or sparse cell counts; check that no cell is implausibly small before treating a phi value as a stable estimate of association strength.