Degrees of freedom

Degrees of freedom count the independent pieces of information remaining in a dataset after parameters have been estimated. Each parameter estimated from the data consumes one piece.

Also known asdf

ByHoang TruongUpdated June 30, 2026

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A two-step waterfall chart starts with a sample of 20 observations, then deducts 1 degree of freedom because the sample mean must be estimated from the same data before the variance can be computed. The result is 19 degrees of freedom, which determines the critical value from the t-distribution — a smaller df produces a heavier-tailed distribution and a more cautious critical value.

Where it fits

SubjectData Analysis & StatisticsCoreTopicHypothesis TestingCore

The formula

LaTeX

\text{df} = n - 1

Variables

$df$: Degrees of freedom (—)
$n$: Sample size (—)

One-sample t-test: n observations, one mean estimated

LaTeX

df = n - k - 1

Variables

$df$: Residual degrees of freedom (—)
$n$: Number of observations (—)
$k$: Number of regressors (slope parameters) (—)

Multiple linear regression: k regressors plus one intercept estimated

Check yourself

PracticeCORE

A researcher tests whether the mean waiting time at a bank branch equals 8 minutes, using a one-sample t-test with 25 observations. How many degrees of freedom does this test use?

Select an answer to check your understanding.