Degrees of Freedom

Degrees of freedom are the maximum number of logically independent values, which may vary in a data sample. Suppose we have two choices of shirt to wear at a party then the degree of freedom is one, now suppose we have to again go to the party and we can not repeat the shirt then the choice of shirt we are left with is One then in this case the degree of freedom is zero as we do not have any choice to choose on the last day. Degrees of freedom are calculated by subtracting one from the number of items within the data sample.

In this article, we will learn about the Degree of freedom, its formula, its example, and others in detail.

Degrees of Freedom

Degrees of Freedom are the maximum number of independent values that can vary in a sample space. Degree of freedom is generally calculated when we subtract one from the given sample of data. Degrees of freedom are very helpful for ensuring the validity of chi-square tests, t-tests, high f-tests, and others. 

These tests are very helpful for deducing the validity of any hypothesis. Degree of Freedom helps us to make various crucial decisions.

Degrees of Freedom Definition

We define Degree of Freedom as choices that any given sample of data has. Suppose you are given to choose an option from four different options and you have to choose one option then the degree of freedom in this case is 3.

Degree of freedom formula is shown in the image shown below. In the image, N is total number of options available to us then degree of freedom is N – 1.

Degrees of Freedom Formula

Degrees of Freedom Example 

Degree of Freedom can easily be understood with the help of the following example. Suppose you have three packets of A, B, and C of food to eat in a day.

• For Breakfast: You can have any of the three packets (A, B, and C), but you choose to eat packet A.
• For Lunch: As, you have only two packets left (B, C) you have two choices and you choose B.
• For Dinner: You have only one packet left i.e. C so you are forced to eat that one only.

So, we have two levels of freedom to choose our food for breakfast in a day, so the degree of freedom in this case is two, but for lunch, we have one freedom so the degree of freedom in this case is one and for dinner, we have no choices so the degree of freedom is one.

What Degrees of Freedom Tells?

Degrees of freedom are the number of independent variables estimated in the statistical analysis of various data sets. The value of the variables of the degree of freedom is without constraint, but they do impose restrictions on other variables for the data set to comply with our estimated parameters.

Initially, some numbers are chosen at random, but data set adds up to mean value.

Degrees of Freedom Formula

Formulas to Calculate Degrees of Freedom with respect to the number of samples,

One-Sample T-Test Formula	dF = n − 1
Two-Sample T-Test Formula	dF = n1 + n2 − 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)

Degrees of Freedom are often discussed in relation to various methods of hypothesis testing in mathematics, such as chi-square. It is important to calculate degrees of freedom when trying to understand the importance of the chi-square arithmetic and the validity of the null hypothesis.

Degree freedom formula is equal to the size of a sample of data minus one.

D_f = N – 1

where,

• D_f is Degree of Freedom
• N is Actual Sample Size

Degree of Freedom and Probability Distributions

Degrees of Freedom are also used to define the probability distributions for the various hypothesis tests. For example, hypothesis tests use the t-distribution, F-distribution, and chi-square distribution for determining the statistical significance of the given data. Probability distributions contain various distributions which use the Degree of Freedom to define the shape of various tests.

Use of Degrees of Freedom

Degree of Freedom is widely used in engineering and mathematics to find the total possible ways to complete a task. It is used in statistics to find the number of ways in which we can complete a task or it represents the total possible variation in the values of the data set. We use the concept of degrees of freedom to validate t-tests and chi-square tests.

Understanding Degrees of Freedom

We can understand the meaning and concept of the degree of freedom by going through the following example.

Suppose we take four marbles that are to be distributed between four children A, B, C, and D in the same order respectively. Now the degrees of freedom they have is explained as,

• For A we have all four choices of marbles available then the degrees of freedom for A is 4-1 = 3
• For B we have all three choices of marbles available then the degrees of freedom for B is 3-1 = 2
• For C we have all two choices of marbles available then the degrees of freedom for C is 2-1 = 1
• For D we have all only one choice of marbles available then the degrees of freedom for D is 1-1 = 0

Degrees of freedom of the given sample of data is given using the formula,

D_f = n – 1

where,

• D_f is Degree of Freedom
• n is Number of Choices

How to Determine Degrees of Freedom?

Degrees of Freedom for a set of data is calculated by subtracting one from the number of items within the set. As all items within the set can be randomly selected until there is only one item remaining.

Chi-Square Tests

Degrees of Freedom are very important to understand the importance of a chi-square statistic and to validate the null hypothesis.

Chi-Square Tests are of two different kinds,

• Test of Independence
• Goodness-of-Fit Test

Degrees of Freedom are utilized for these tests to determine whether a certain null hypothesis can be accepted or rejected based on the total number of variables and samples within the experiment space.

T-Test

Before performing a t-test, we first calculate the value of t for the given sample of data and compare it to the critical value. The critical value is determined by using the given data set’s t distribution with the correct degrees of freedom.

It is concluded that sets with lower degrees of freedom have a higher chance of extreme values, while the sets with higher degrees of freedom are much closer to a normal distribution curve.

Application of Degrees of Freedom

Various applications of Degrees of Freedom are discussed below

• Degrees of Freedom in statistic is used to define shape of t-distribution used in t-tests while calculating p-value. 
• Different degrees of freedom will display different t-distributions depending on sample size.
• Degrees of Freedom calculations also help us to understand the importance of a chi-square statistic and validity of null hypothesis.

Degrees of freedom also have various applications outside the field of statistics.

For example, take a company that has to decide on how many raw materials to acquire for its manufacturing process. The company has to consider two values within this data set,

• Number of Raw Materials to Acquire
• Total cost of Raw Materials

Company is free to decide on one of the two items, but their choice will affect outcome of other value. When company fixes number of raw materials to acquire, cost of total material is not controlled by the company. Whereas when company decided to limit total amount to spend, company is not able to control quantity of raw material it purchases. 

Here the company is only free to choose any one of two values, hence degree of freedom is one in above example.

Examples on Degree of Freedom

Example 1: Determine the Degree of Freedom for the given set of data.

Data: 5, 7, 4, 6, 10, 12

Solution: 

Number of Values, n = 6

Degree of Freedom = n-1 

= 6 – 1

= 5

Example 2: Evaluate the Degree of Freedom for the given set of observations.

Observations: 1, 7, 5, 12, 17, 18, 19, 25

Solution: 

Number of Values, n = 8

Degree of Freedom = n – 1 

= 8 – 1

= 7

Example 3: Evaluate Degree of Freedom for the given set of observations.

• Observation 1: 1, 7, 5, 12, 17, 18, 19, 25
• Observation 2: 14, 15, 21, 29, 10

Solution: 

• Number of Values in Observation 1, n₁ = 8
• Number of Values in Observation 2, n₂ = 5

Degree of Freedom = n₁ + n₂ – 2 

= 8 + 5 – 2 

= 11

Example 4: Evaluate the Degree of Freedom for the given set of observations.

• Observation 1: 1, 6, 5, 13, 17
• Observation 2: 12, 11, 26

Solution: 

• Number of values in observation 1, n₁ = 5
• Number of values in observation 2, n₂ = 3

Degree of Freedom = n₁ + n₂ – 2 

= 5 + 3 – 2 

= 6

FAQs on Degrees of Freedom

What is Degree of Freedom?

Degree of freedom is the number of independent values of the sample in the given data set minus one. It is the total choice available to us in a given sample space. For example, if we have to choose from 5 different choices then the degree of freedom in that case is 4 (5-1).

How to Determine Degrees of Freedom?

Degrees of freedom of any sample space are calculated by subtracting one from the total number of values of the set.

What is Degree of Freedom Formula?

Degrees of freedom of given sample of data is calculated using the formula, D_f = n – 1

What is Degree of Freedom of Diatomic Gas?

Degree of freedom of a diatomic gas is 5. It has 3 degrees of freedom from the translatory motion in the three-axis and 2 degrees of freedom from its rotational motion.