Degrees of freedom: what it means and how to read it

In the report: df — The degrees of freedom of the statistic next to it: how much independent information the estimate rests on.

How to read it

Degrees of freedom — roughly, how much data is left to estimate the spread once the means are estimated. You do not interpret them; you report them next to the statistic, for example t(45) = 2.10.

In plain words

A student measured the height of 50 pupils and wrote a short report. Her teacher asked her to add the degrees of freedom, and she found 49. This number says roughly how much information is left over once the average has been worked out. She does not have to judge it or explain it. She only writes it in its usual place next to the main result, so readers can see how much information stood behind it. With 10 pupils the same report would show 9, and readers would know the result rests on much less.

Example

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

— the sample data (50 rows) the examples use
— =STATS_LINEAR(Data!A1:T51, "y", "x1,x2", 0)
— two steps; pass is coded 0 and 1 =STATS_LOGISTIC(Data!A1:T51, "pass", "x1,x2|x", 0, 50, 1e-8)
— =STATS_PEARSON(Data!A1:T51, "x,y,x1",0.95,"two-sided")

Where you see it

STATS_LINEAR — Multiple linear regression (OLS) with hierarchical steps separated by | and interactions a*b.
STATS_LOGISTIC — Binary logistic regression with optional ridge regularization (lambda >= 0).
STATS_PEARSON — Pearson correlation (r).
STATS_SPEARMAN — Spearman rank correlation (rho): the correlation for rating scales, ranks and skewed data — does one go up as the other goes up?
STATS_PARTIAL — Partial correlation: the Pearson correlation of two variables with the controls held constant (SPSS PARTIAL CORR). Two variables give one correlation, three or more a matrix, as STATS_PEARSON.
STATS_CHI2 — Pearson chi-square test for association between two categorical columns.
STATS_CRAMER — Cramer's V effect size for association between two categorical columns.
STATS_NORMALITY — Shapiro–Wilk test of normality for every selected column — the Shapiro–Wilk part of the SPSS EXAMINE table "Tests of Normality" (without its Kolmogorov–Smirnov test).
STATS_PAIRED — Paired-samples t-test for two related measurements (e.g., pre vs post).
STATS_INDEPENDENT — Independent-samples t-test for two groups: the two selected columns are the groups.
STATS_ANOVA — One-way ANOVA: each selected column is a group. Classic (Student) or Welch ANOVA, chosen by welch (auto by default), with homogeneity of variances and post-hoc comparisons.
STATS_KRUSKAL — Kruskal–Wallis test for two or more independent groups (usually three or more; for two it agrees with STATS_MANN_WHITNEY) — the rank replacement for STATS_ANOVA when the data are not normal, or are ranks or rating scales. Each selected column is a group, as in STATS_ANOVA; no column is dropped.
STATS_ONE_SAMPLE — One-sample t-test against mu0.
STATS_CI — Calculates the confidence interval of the mean for filtered columns (Student's t).
STATS_ANALYZE — Runs several analyses on one range and returns one report.

t statistic — A difference measured in standard errors; its sign is the sign of the difference. Its p comes from the t distribution with the degrees of freedom next to it.
Likelihood-ratio chi-square — How much −2 Log Likelihood dropped compared with the model with the intercept only.
Wald statistic — The Wald statistic of the coefficient: (B / StdError)². Empty when lambda is above 0.
F change — The F test of R Square Change: whether the predictors added at this step explain more than chance. At the first step it equals the F of the model. Empty when the step entered no predictor (df1 = 0).
z statistic — The test statistic on the normal scale.

Get it in your sheet

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

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