Degrees of Freedom (One-Sample t)

Degrees of freedom count how many values in a calculation are genuinely free to vary. Once you have used the data to compute the sample mean, the last observation is pinned: if four of five numbers are known and the mean is known, the fifth is forced. So estimating a spread around your own sample mean leaves n − 1 independent pieces of information, and that is the row you use in the t-table. A sample of 25 gives df = 24 and a two-sided 5% critical value of 2.064, noticeably wider than the normal 1.96.

The practical trap is grabbing the wrong row. df = n − 1 is the one-sample and paired case; a pooled two-sample t uses n₁ + n₂ − 2, and a chi-square test of independence uses (rows − 1)(columns − 1). Ronald Fisher gave the concept its name and its geometry in the 1920s, picturing the data as a point in n-dimensional space and each estimated parameter as one dimension spent. As n grows the distinction fades — by df = 120 the t critical value is 1.980, within one percent of the normal — which is why large-sample work stops worrying about it.