Degrees of Freedom

What is Degrees of Freedom?

Degrees of freedom definition is a mathematical equation used principally in statistics, but also in physics, mechanics, and chemistry. In a statistical calculation, the degrees of freedom illustrates the number of values involved in a calculation that has the freedom to vary. The degrees of freedom can be computed to ensure the statistical validity of t-tests, chi-square tests and even the more elaborated f-tests.

In this lesson, we will explore how degrees of freedom can be used in statistics to identify if outcomes are significant.

Use of Degrees of Freedom

Tests like t-tests, chi-square tests are frequently used to compare observed data with data that would be anticipated to be obtained as per a particular hypothesis.

Degrees of Freedom Example

Examples of how degrees of freedom can enter statistical calculations are the t-tests and chi-squared tests. There are a number of t-tests and chi-square tests which can be differentiated with the help of degrees of freedom.

Let’s consider a degree of freedom example. Suppose a medicinal trial is carried out on a group of patients and it is postulated that the patients receiving the medication would display increased heartbeat in comparison to those that did not receive the medication. The outcome of the test could then be evaluated to identify whether the difference in heart rates is considered crucial, and degrees of freedom are part of the computations.

Degrees of Freedom Formula

The statistical formula to find out how many degrees of freedom are there is quite simple. It implies that degrees of freedom is equivalent to the number of values in a data set minus 1, and appears like this:

df = N-1

Where, N represents the number of values in the data set (sample size).

That being said, let’s have a look at the sample calculation.

If there is a data set of 6, (N=6).

Call the data set X and make a list with the values for each data.

For this example data, set X of the sample size includes: 10, 30, 15, 25, 45, and 55

This data set has a mean, or average of 30. Find out the mean by adding the values and dividing by N:

(10 + 30 + 15 + 25 + 45 + 55)/6= 30

Using the formula, the degrees of freedom will be computed as df = N-1:

In this example, it appears, df = 6-1 = 5

This further implies that, in this data set (sample size), five numbers contain the freedom to vary as long as the mean remains 30.

Critical Values

Having the awareness of the degrees of freedom for a sample or the population size does not provide us a whole lot of substantial information by itself. This is because after we perform computations of the degrees of freedom, which are actually the number of values in a calculation that we can vary, it is essential to look up the critical values for our equation with the help of a critical value table. Note that these tables can be found online or in textbooks. When using a critical value table, the values found in the table identify the statistical significance of the outcomes.

Solved Examples on Finding How Many Degrees of Freedom

Now that we know the degree of freedom meaning, let's get to learn how to find the degrees of freedom.


Evaluate the Degree of Freedom For a Given Sample or Sequence: x = 3, 6, 2, 8, 4, 2, 9, 5, 7, 12


Given n= 10


DF = n-1

DF = 10-1

DF = 9


Determine the Degree of Freedom For the Sequence Given Below:

x = 12, 15, 17, 25, 19, 26, 35, 46

y = 18, 32, 21, 43, 22, 11


Given: n1 = 8 n2 = 6

Here, there are 2 sequences, so we require to apply DF = n1 + n2 – 2

DF = 8 + 6 -2

DF = 12

Fun Facts

  • Since degrees of freedom calculations determine the number of values in the final calculation, they are allowed to vary, and to even contribute to the validity of a result.

  • Degree of freedom calculations are typically dependent upon the sample size, or observations, and the criterions to be estimated, but usually, degree of freedom mathematics and statistics equals the number of observations minus the number of criterion/parameter.

  • There will be more degrees of freedom with a larger size of sample.

FAQs (Frequently Asked Questions)

1. What are the Formulas of Calculating the Degrees of Freedom?

Answer: There are several formulas to calculate degrees of freedom with respect to sample size. Following are the formulas to calculate degrees of freedom based on sample:

  • One Sample T Test Formula : DF = n−1

  • Two Sample T Test Formula: DF = n1 + n− 2

  • Simple Linear Regression Formula: DF = n−2

  • Chi Square Goodness of Fit Test Formula: DF = k−1

  • Chi Square Test for Homogeneity Formula: DF = (r−1)(c−1)

2. How are Degrees of Freedom Used in Standard Deviation?

Answer: Another place where degrees of freedom occur is in the standard deviation formula. This appearance is not as clear and apparent, but we can notice it if we know where to look. To determine a standard deviation we are looking at the "average" deviation from the mean. However, after we do subtraction of the mean from each data value and squaring the differences, we end up dividing by n-1 instead of n as we might anticipate.

The occurrence of the n-1 takes place from the number of degrees of freedom. Because the sample mean and the n data values are being used in the formula, there are n-1 degrees of freedom.

3. How are Degrees of Freedom Used in Advanced Statistical Techniques?

Answer: More advanced statistical techniques apply more complex ways of counting the degrees of freedom. When computing the test statistic for two means having independent samples of n1 and n2 elements, the number of degrees of freedom consists of a little complicated formula. It can be calculated using the smaller of n1-1 and n2-1.

Another example of counting the degrees of freedom shows up with an F test. In carrying out an F test we have k samples each of size n—the degrees of freedom in the numerator will be k-1 and in the denominator will be k(n-1).