Phi: what it means and how to read it
In the report: Phi — The strength of the relation, from the chi-square: for a 2 × 2 table with the sign of the relation (the sign depends on the order of the categories), for a larger table it equals the square root of χ² / N.
How to read it
For a 2 × 2 table it reads like a correlation of two yes/no answers: 0.1 small, 0.3 medium, 0.5 large. For larger tables report Cramér's V instead.
In plain words
A teacher asked 100 students two questions: Are you a girl? Did you join a club? Half were girls, and half joined. How strongly do the two answers go together? Phi came out 0.32: 33 of the 50 girls joined but only 17 of the 50 boys. If she guesses "girls join, boys do not", she is right for 66 students out of 100, the 33 girls who joined and the 33 boys who did not, where tossing a coin would be right for about 50. Reports call this a medium link. Its sign only shows which answer she put first. For a table with more than two rows or columns she would use Cramér's V instead.
How it is computed
2 × 2: φ = (ad − bc) / √((a+b)(c+d)(a+c)(b+d)); larger: φ = √(χ² / N)Example
The cell with Phi is selected in each report; point at a label to see its note.
- — the sample data (50 rows) the examples use
- — with the Chi-Square Tests and Symmetric Measures tables at the end
=STATS_CHI2(Data!A1:T51, "gender", "group", TRUE)
Where you see it
STATS_CHI2— Pearson chi-square test for association between two categorical columns.
Related values
- Cramér's V — The strength of the relation of two categorical variables, from 0 (none) to 1 (one determines the other).
- Eta squared (η²) — The share of the total variance explained by the groups: 0 — none, 1 — all.
- Omega squared (ω²) — The share of variance explained by the groups, corrected for sample size: a less biased η², slightly smaller. Can be below 0 when there is no effect.
- McFadden pseudo R² — How much the model improves the log-likelihood over the model with the intercept only.
- Cox & Snell pseudo R² — A pseudo R² from the likelihood ratio; even a perfect model stays below 1.
Get it in your sheet
One formula in a cell gives the whole report, Phi included. ALS Stats is a free add-on for Google Sheets™.
Install ALS Stats from Google Workspace Marketplace (opens in a new tab)