What Are Degrees of Freedom?
Degrees of freedom (df) represent the number of independent values that can vary in a statistical calculation. When computing a sample mean from n observations, one degree of freedom is lost because the values must sum to a specific total, leaving n-1 free to vary. This concept is fundamental to t-tests, chi-square tests, F-tests, and ANOVA.
Understanding df is crucial because it determines the shape of the sampling distribution used for hypothesis testing. More degrees of freedom produce distributions closer to the standard normal, while fewer df produce heavier-tailed distributions requiring larger critical values.
Formulas by Test
| Test | Formula | Example |
|---|---|---|
| One-sample t | n - 1 | n=25: df=24 |
| Two-sample t | n1 + n2 - 2 | n1=25,n2=30: df=53 |
| Paired t | n - 1 | n=20 pairs: df=19 |
| Chi-square | (r-1)(c-1) | 3x4 table: df=6 |
| One-way ANOVA | Between: k-1; Within: N-k | k=3,N=60: 2,57 |
Common df and t-Critical Values (alpha=0.05, two-tailed)
| df | t-critical |
|---|---|
| 10 | 2.228 |
| 20 | 2.086 |
| 30 | 2.042 |
| 60 | 2.000 |
| 120 | 1.980 |
Why df Matters
Degrees of freedom affect the width of confidence intervals, the critical values for hypothesis tests, and the power of statistical tests. Using incorrect df can lead to either inflated Type I error rates or unnecessarily conservative tests.
Frequently Asked Questions
Can df be a decimal?
Yes, in Welch's t-test for unequal variances, df is calculated using the Satterthwaite approximation and can produce non-integer values. The result is typically rounded down for table lookup.
What happens as df increases?
As df increases, the t-distribution approaches the standard normal distribution. Beyond df=120, the t and z distributions are nearly identical.
Why do we lose degrees of freedom?
Each parameter estimated from data constrains the remaining values. Estimating a mean from n values leaves n-1 free to vary. Estimating both mean and variance from n values loses 1 df for the mean.