Adjusted residuals: what it means and how to read it

In the report: Adjusted residuals — With details = TRUE: the adjusted standardized residual of each cell (SPSS "Adjusted Residual"): how far the count is from the one expected if the two columns were independent, in standard errors. Empty when there is one row or one column.

How to read it

Beyond ±2 the cell has clearly more (positive) or fewer (negative) rows than independence would give; these cells are where a significant STATS_CHI2 comes from.

In plain words

A teacher asked 100 students which club they joined and switched on details. She wanted to know which cells of the table stand out. In the cell for girls in the chess club she expected 8 girls if boys and girls chose in the same way. She counted 14. The adjusted residual for that cell came out 2.6. Above 2 means many more students than expected, and below minus 2 many fewer. Now she can say that more girls joined chess than expected, and that this cell is where the link in her table comes from. The other cells, near 0, look as expected.

How it is computed

(Oᵢⱼ − Eᵢⱼ) / √(Eᵢⱼ · (1 − Rᵢ/n) · (1 − Cⱼ/n)), Eᵢⱼ = Rᵢ · Cⱼ / n

Example

The cell with Adjusted residuals is selected in each report; point at a label to see its note.

— the sample data (50 rows) the examples use
— with row and column percentages and adjusted residuals =STATS_CROSSTABS(Data!A1:T51, "gender", "group", TRUE)

Where you see it

STATS_CROSSTABS — Observed contingency table for a pair of categorical columns.

Chi-Square Tests — With details = TRUE: the tests of the table as in SPSS CROSSTABS /STATISTICS=CHISQ, at the very end of the report. Pearson Chi-Square, Likelihood Ratio and N of Valid Cases are always there; Continuity Correction and Fisher's Exact Test only for a 2 × 2 table; Linear-by-Linear Association only when both columns have codes (numbers, or labels with numbers).
Symmetric Measures — With details = TRUE: the strength of the relation, Phi and Cramér's V, as in SPSS CROSSTABS /STATISTICS=PHI. Follows the Chi-Square Tests table.
N of Valid Cases — The rows the test is computed on: both answers present and applicable.
Observed counts — The crosstab of the counts the test uses — the same table as STATS_CROSSTABS.
Expected counts — The counts expected if the two variables were unrelated: row total × column total / N, for the non-empty categories. The test compares them with the observed counts.

Get it in your sheet

One formula in a cell gives the whole report, Adjusted residuals included. ALS Stats is a free add-on for Google Sheets™.

Install ALS Stats from Google Workspace Marketplace (opens in a new tab)