Degrees of freedom calculator

What are degrees of freedom?

Degrees of freedom is the count of values in a calculation that are still free to vary. The University of Texas Statistical Online Services puts it as the number of values which are free to vary, and the seat analogy they use makes it concrete: in a room with 30 chairs, 29 people can sit wherever they like, and the last person has no choice left.

The same thing happens once you estimate something from your data. Say five numbers have a mean of 20, so they add up to 100. You can set the first four to anything at all, say 14, 22, 31 and 18. Those add to 85, so the fifth number has to be 15. Four values were free, one was forced, and the degrees of freedom is 4.

That is the rule behind every formula on this page. Start with how many numbers you have, then subtract one for each quantity you estimated from those same numbers. In statistics output it is nearly always written df, and occasionally as the Greek letter nu. So df in statistics and degrees of freedom mean the same thing.

Degrees of freedom formula

For a single sample the formula is:

df = n − 1

n is the number of observations, and the 1 is the sample mean, which you had to work out before you could measure anything about spread. This is the same n − 1 that appears in the denominator of the sample variance, which is why our variance calculator divides by n − 1 for a sample and by N for a population.

Every other test follows the same pattern with a different count of estimates. A two-sample t-test estimates two means, so it costs two:

df = n₁ + n₂ − 2

A contingency table fixes its row and column totals, so the last row and last column are forced:

df = (r − 1)(c − 1)

How to find degrees of freedom

Three questions get you there for any test:

1. Count the values you actually have. For a paired test that means pairs, not measurements. For ANOVA it means every observation across all groups.
2. Count what the test estimates from those same values. A t-test estimates means. A regression estimates a slope for each predictor plus an intercept.
3. Subtract the second number from the first.

Worked example

You are comparing test scores for two classes. Class A has 12 students, class B has 15, and you are willing to assume the two groups have similar spread, so you run a pooled two-sample t-test.

• Total observations: 12 + 15 = 27
• Estimates: two group means
• Degrees of freedom: 27 − 2 = 25

Look 25 df up in a t table at the 0.05 level, two-tailed, and the critical value is 2.0595. If your t statistic comes out beyond ±2.0595 the difference is significant at that level. Select the same test in the calculator above and you get the df, the critical value and a slice of the t table around 25 df without opening a textbook.

Degrees of freedom for a t-test

Which formula you need depends on how many samples you have and what you are assuming about their variances.

• One-sample or paired t-test: df = n − 1. For the paired version, n is the number of pairs. Twelve people measured before and after gives n = 12, not 24.
• Two-sample t-test with equal variances: df = n₁ + n₂ − 2. This is the pooled test, and it assumes both groups share one variance.
• Two-sample t-test with unequal variances: the Welch-Satterthwaite formula, which mixes the two sample sizes and the two variances.

The Welch formula

df = (s₁²/n₁ + s₂²/n₂)² / [ (s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1) ]

Take the two classes again, but now class A has a standard deviation of 8.60 and class B has 16.03. The pooled formula would still say 25 df. Welch gives 22.22, because the wider spread in the larger group costs you precision.

That decimal is not a mistake. R, SPSS and Excel all use the exact value. Penn State’s STAT 415 notes say to take the integer portion when you are reading a printed table, so 22.22 becomes 22. The critical value barely moves either way: 2.0727 at 22.22 df against 2.0739 at 22.

Degrees of freedom for a chi-square test

Chi-square has two common tests and they use different formulas.

For a test of independence on a contingency table, df = (r − 1)(c − 1). A 3 by 4 table has 2 × 3 = 6 degrees of freedom. The row and column totals come from the data, so once you fill in a 2 by 3 block of cells the rest of the table is determined.

For a goodness of fit test, df = k − 1 − m, where k is the number of categories and m is how many parameters you estimated from the same data. Comparing six dice faces against an even distribution gives df = 6 − 1 = 5, because you estimated nothing. Fitting a Poisson distribution to six categories means estimating the rate from your data first, so m = 1 and df drops to 4.

People forget the m term more than anything else on this page. Skip it and your df is too high, your critical value is too big, and you end up failing to reject a null hypothesis you should have rejected.

Degrees of freedom in ANOVA and F tests

An F test uses two degrees of freedom, one for the numerator and one for the denominator, and you need both to read the table.

One-way ANOVA with k groups and N observations in total:

• Between groups (the numerator): df = k − 1
• Within groups (the denominator): df = N − k
• Total: df = N − 1

Forty students split across four teaching methods gives 3 and 36. At the 0.05 level the critical value of F(3, 36) is 2.8663. The two parts always add back to N − 1, which is a quick way to check you have not miscounted, and the sums of squares they divide into split the same way.

Two-way ANOVA splits the same total further. With a levels of factor A, b levels of factor B and n observations per cell, you get a − 1 for factor A, b − 1 for factor B, (a − 1)(b − 1) for the interaction, and ab(n − 1) for the error term. All three F tests share that error df as their denominator.

An F test comparing two variances is simpler: df₁ = n₁ − 1 and df₂ = n₂ − 1, with the larger variance on top. F tables are one-tailed, so for a two-sided test at 0.05 you look up 0.025.

Degrees of freedom in regression and correlation

A straight line through your data estimates two things, a slope and an intercept, so simple linear regression leaves df = n − 2. Thirty observations gives 28 df, and a t test on the slope uses a critical value of 2.0484 at the 0.05 level. An exponential regression fitted by taking logs carries the same n – 2, because underneath it is still a straight line with two parameters.

Add predictors and each one costs another degree of freedom. With p predictors plus an intercept, the residual df is n − p − 1. Sixty observations and four predictors leaves 55. The overall F test for that model is F(4, 55), while a t test on any single coefficient uses the 55.

This is why a model with too many predictors for its sample size stops being testable. Fit ten predictors to twelve observations and you are down to a single residual df, which makes every standard error enormous and every coefficient insignificant.

Testing a Pearson correlation uses df = n − 2 for the same reason, since r depends on both the x mean and the y mean.

Degrees of freedom table by test

Every formula the calculator uses, in one place.

The calculator prints the critical values alongside the df, which is the part a printed table makes you look up separately.

Why degrees of freedom change your answer

Degrees of freedom pick which row of the table you read, and with small samples that row matters a lot. At the 0.05 level, two-tailed, the critical t value is:

• 5 df: 2.5706
• 10 df: 2.2281
• 30 df: 2.0423
• 100 df: 1.9840
• 1,000 df: 1.9623

With 5 df your test statistic has to reach 2.5706 to count as significant. With 100 df, 1.9840 is enough. Small samples have fatter tails because a variance estimated from five numbers is itself unreliable, and the t distribution widens to account for that. Past about 120 df the difference stops mattering and t is close enough to the normal distribution’s 1.96.

Getting the df wrong by one or two is usually harmless with a large sample and can flip the conclusion with a small one. The gap between 5 and 6 df at the 0.05 level is about 0.13, which is enough to decide a borderline test.