How to Calculate Degrees of Freedom

The degrees of freedom (df) play a crucial role in several types of statistical analysis. The exact way to calculate degrees of freedom depends on the specific analysis you are conducting. Below, we show how to calculate degrees of freedom for several common t-test scenarios and for the chi-square test for independence.

One-Sample t-Test

In a one-sample t-test, you're comparing the sample mean to a known population mean.

Formula for degrees of freedom (df):
df = n - 1
where n is sample size.
If sample size is 30, degrees of freedom would be:
df = 30 - 1 = 29

Independent Two-Sample t-Test

In an independent two-sample t-test, you're comparing the means of two independent groups. The degrees of freedom depend on whether the two samples have equal or unequal variances and on the precision required.

Equal Variances Assumed (Pooled t-test)

If you assume equal variances between the two groups, the degrees of freedom are calculated using a pooled variance estimate.

Formula for degrees of freedom (df):
df = n₁ + n₂ - 2
where n₁ is sample size in the first group, and n₂ is sample size in the second group.
If you have 30 observations in the first group (n₁ = 30) and 40 observations in the second group (n₂ = 40), the degrees of freedom would be:
df = 30 + 40 - 2 = 68

Unequal Variances Assumed (Welch’s t-test)

If you do not assume equal variances between the two groups, then Welch’s t-test is used; and the degrees of freedom are calculated using a more complex formula that accounts for the difference in variances between the two groups.

Formula for degrees of freedom (Welch-Satterthwaite equation):
num = (s₁²/n₁ + s₂²/n₂)²

den = [(s₁²/n₁)²/(n₁ - 1)] + [(s₂²/n₂)²/(n₂ - 1)]

df = num / den
where s₁² and s₂² are sample variances, and n₁ and n₂ are sample sizes in the two groups.
This formula is complex and requires you to know both the sample variances and sizes. If you have unequal variances, you'll use this formula to calculate the degrees of freedom. The result is typically a non-integer value, and you would round it to the nearest whole number for practical use.

Unequal Variances Assumed (Conservative Approach)

Compared to the Welch-Satterthwaite equation, the conservative approach simplifies the calculation; but it results in wider confidence intervals and more cautious hypothesis tests.

Formula for degrees of freedom (conservative approach):
df = min(n₁ - 1, n₂ - 1 )
where n₁ and n₂ are sample sizes in the two groups.
If you have 20 observations in the first group (n₁ = 20) and 35 observations in the second group (n₂ = 35), the degrees of freedom would be:
df = 20 - 1 = 19

Paired t-Test

In a paired t-test, you're comparing the means of two related groups or measurements (e.g., before and after treatment on the same subjects). The degrees of freedom are based on the number of paired differences.

Formula for degrees of freedom:
df = n - 1
where n is the number of pairs of observations (or the number of differences).
If you have 15 pairs of data points (e.g., 15 subjects before and after treatment), the degrees of freedom would be:
df = 15 - 1 = 14

Chi-Square Test of Independence

The test of independence is used to determine whether two categorical variables are independent of each other. This test is often applied to contingency tables (cross-tabulations of two categorical variables).

Formula for degrees of freedom:
df = (r - 1)(c - 1)
where r is the number of rows in the contingency table, and c is the number of columns.
The degrees of freedom are calculated based on the number of categories (levels) in each of the two variables you're testing. The reasoning is that the more rows and columns there are, the more constraints there are on how the observed and expected frequencies can vary.