When computing the degrees of freedom for ANOVA How is the within group estimate calculated?

Created by Rita Rain

Reviewed by Dominik Czernia, PhD candidate and Jack Bowater

Last updated: Oct 01, 2022

This degrees of freedom calculator will help you determine this crucial variable for one-sample and two-sample t-tests, chi-square tests, and ANOVA**. Read the text to find out:

• What degree of freedom is (degrees of freedom definition);
• How to find degrees of freedom; and
• The degrees of freedom formula.

What are degrees of freedom? Definition

Let's start with a definition for degrees of freedom:

Degrees of freedom indicate the number of independent pieces of information used to calculate a statistic; in other words - they are the number of values that are able to be changed in a data set.

That may sound too theoretical, so let's take a look at an example:

Imagine we have two numbers: x, y, and the mean of those numbers: m. In this data set of three variables, how many degrees of freedom do we have? The answer is 2. Why? Because 2 is the number of values that can change. If you choose the values of any two variables, the third one is already determined. Look:

• If x equals 2 and y equals 4, you can't pick any mean you like; it's already determined:

  m = (x + y) / 2

  m = (2 + 4) / 2

  m = 3.

• If you assign 3 to x and 6 to m, then y's value is "automatically" set - it's not free to change, because:

  m = (x + y) / 2

  6 = (3 + y) / 2

  12 = 3 + y

  12 - 3 = y

  y = 9

Any time you assign some two values, the third has no "freedom to change", hence there are two degrees of freedom in our scenario.

Now that we know what degrees of freedom are, let's learn out how to find df.

How to find degrees of freedom - formulas

The formula for degrees of freedom depends on the type of statistical test you're performing. Below, you'll see equations for the most popular ones:

1. 1-sample t-test:

   df=N−1\textrm{df} = N - 1,

   where:

   • df - Degrees of freedom; and

   • N - Total number of subjects/values.

2. 2-sample t-test (samples with equal variances):

   df=N1+N2−2\textrm{df} = N_1 + N_2 - 2,

   where:

   • N₁ - Number of values from the first sample; and

   • N₂ - Number of values from the second sample.

3. 2-sample t-test with unequal variances (Welch’s t-test):

   In this case, we calculate an approximation of the degrees of freedom:

   df=(Var1N1 +Var2N2)2Var12N12(N1−1)+Var22 N22(N2−1)\textrm{df} = \frac{\left(\frac{\textrm{Var}_1}{N_1}+ \frac{\textrm{Var}_2}{N_2}\right)^2}{\frac{\textrm{Var}_1^2}{N_1^2 (N_1-1)}+\frac{\textrm{Var}_2^2}{N_2^2 (N_2-1)}}

   where Var - Variance.

✅ As you can see, the number of values in samples heavily influences the number of degrees of freedom. Learn more with our sample size calculator. Or, if you just wish to perform a t-test quickly and without worrying about df, use Omni's t-test calculator - it will take care of everything!

1. ANOVA:

   • Degrees of freedom between groups:

     dfbetween=k−1\textrm{df}_{\rm between} = k - 1,

     where k - Number of groups or cell means.

   • Degrees of freedom within groups:

     dfwithin =N−k\textrm{df}_{\rm within} = N - k

   • Total degrees of freedom:

     dftotal =N−1\textrm{df}_{\rm total} = N - 1

2. Chi-squared test of independence

   df=(rows−1)⋅(columns−1)\textrm{df} = (\textrm{rows} - 1) \cdot (\textrm{columns} - 1)

You can discover more about computing χ² with our dedicated chi squared calculator.

If you're wondering how to find df quickly - use our degrees of freedom calculator. It includes all of the above formulas.

Degrees of freedom calculator

This is how to use the df calculator:

1. Choose the statistical test you're using.

2. Enter the variables which will appear in the rows below, e.g. the sample size.

3. You'll find the result in the last box of the df calculator.

FAQ

How to calculate degrees of freedom for t-test?

To calculate degrees of freedom of a 1-sample t-test:

1. Determine the size of your sample (N).

2. Subtract 1.

3. The result is the number of degrees of freedom.

How to calculate degrees of freedom for chi-square?

To calculate degrees of freedom for the chi-square test, use the following formula:

df = (rows - 1) * (columns - 1),

that is:

1. Count the number of rows in the chi-square table and subtract one.

2. Count the number of columns and subtract one.

3. Multiply the number from step 1 by the number from step 2.

How to calculate degrees of freedom for two-sample t-test?

To calculate degrees of freedom for two-sample t-test, use the following formula:

df = N₁ + N₂ - 2,

that is:

1. Determine the sizes of your two samples.

2. Add them up.

3. Add 2 to the result from the previous step.

How to calculate degrees of freedom for ANOVA?

To calculate degrees of freedom for ANOVA:

1. Subtract 1 from the number of groups to find degrees of freedom between groups.

2. Subtract the number of groups from the total number of subjects to find degrees of freedom within groups.

3. Subtract 1 from the total number of subjects (values) to find total degrees of freedom.

Can degrees of freedom be 0?

Yes, theoretically degrees of freedom can equal 0. It would mean there's one piece of data with no "freedom" to vary and no unknown variables. However, in practice, you shouldn't have 0 degrees of freedom when performing statistical tests.

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