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Degrees of Freedom

Last updated date: 18th Apr 2024
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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 illustrate 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.


The degrees of freedom that are mathematical concepts to statistical calculation represents the number of variables that have the freedom to vary in a calculation. Calculating degrees of freedom can help ensure the validity of chi-square test statistics, t-tests, and highly f-tests, among other tests. These tests are often used to compare data that has been detected with data that would be expected if a particular hypothesis were true.

The fact that the statistical degrees of freedom indicating the number of values ​​in the final calculation is allowed to vary means that they can contribute to the validity of the result. Although the number of observations and parameters to be measured depends on the size of the sample, or the number of observations, and the parameters to be measured, the degree of freedom in the calculations is usually equal to the value of the observations minus the number of parameters. This means that for larger sample size, there are degrees of freedom available.

For Example

Mention that you have seven shirts that you can wear for a week, and you decide to wear each shirt only once a week.

On Sunday, Consider choosing 1 of the 7 shirts. Wear any of the 7 shirts. On the second day, the shirt worn on the first day cannot be selected, and you should choose from the remaining shirts. The pattern continues as follows:

  1. Sunday: 7 shirts to choose from

  2. Monday: 6 shirts to choose from

  3. Tuesday: 5 shirts to choose from

  4. Wednesday: 4 shirts to choose from

  5. Thursday: 3 shirts to choose from

  6. Friday: 2 shirts to choose from

  7. Saturday: 1 shirt to choose from

On the last day, Saturday, there is only one shirt to choose from, which means, in fact, there is no choice. Put it in different names, you are forced on Saturday by your choice of which shirt to wear. In this one week, you have to choose one shirt a day, you have six free days to choose a shirt. It is the same as saying that your choice of shirt is restricted for one day. So, this week, there are six levels of freedom.

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 that 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.

Understanding the Degrees of Freedom

An easy way to understand the degrees of mental freedom is by using an example:

  • Consider a sample of data that combines, in order to simplify, five positive numbers. Values ​​can be any number that does not have a known relationship between them. This data sample, theoretically, can have up to five degrees of freedom.

  • The four numbers in the sample are {3, 8, 5, and 4} and the total number of data samples is expressed as 6.

  • This should mean that the fifth number should be 10. It can't be any other. It does not have the freedom to be different.

  • So the freedom degrees of this data sample are 4.

The free degree formula is equal to the size of a sample of data except one:

\[D_{f} = N-1 \]

Where as;\[D_{f}\]=Degrees of Freedom

N= Actual Sample size

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.

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:

\[d_{f} = N-1\]

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

The Most Important Principles

Knowing the degrees of public freedom or sample does not provide us with much useful information in itself, however. This is because, after calculating the degrees of freedom, which is the value of a fixed number, it is necessary to look at the values ​​of our equation using the value table, which we will find. discuss later. If you look in textbooks or online, you will find these tables. When using a value-based table, the values ​​in the table are used to determine whether the results are statistically significant.

Double chi tests and t-tests are two examples of how degrees of freedom can be included in mathematical calculations. There are several different types of t-tests and chi-square tests that can be divided by the number of degrees of freedom used.

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 criteria to be estimated, but usually, degree of freedom mathematics and statistics equals the number of observations minus the number of criteria/parameters.

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

Application of the Degree of Freedom

Although the level of freedom is a vague and often overlooked concept in mathematics, it is very effective in the real world.

For example, business owners who want to hire employees to produce a product face two changes - function and effect. Additionally, the relationship between employees and output (i.e., the amount of product that an employee can produce) is a liability.

In such a case, the business owners may determine the amount of product to be produced, which may determine the number of employees to be employed, or the number of employees, which may be sufficient for the product to be produced. So, in terms of output and staff, owners have one level of freedom.

FAQs on Degrees of Freedom

1. What is the Level of Freedom with Examples?

The degree of freedom of measurement is the number of independent pieces of information that are included in the calculation of measurement. It is not at all the same as the number of items in the sample. ... You can use 4 people, give 3 degrees of freedom (4 - 1 = 3), or you can use 100 people with df = 99.

2. What is the Liberal Degree Formula?

The most frequently encountered statistics for determining degrees of mathematical freedom are df = N-1. Use this number to view key mathematical values ​​using the critical value table, which also determines the significance of the result statistics.

3. What is the Standard of Freedom used for?

Degrees of freedom refers to the maximum number of logically independent values, which are numerical freedom of variance, in sample data. Degrees of freedom are often discussed in relation to various methods of hypothesis testing in mathematics, such as chi-square.

4. What is the Highest Level of Freedom?

The number of variables needed to define a body movement is called the level of freedom. For example, think of a body in space, with rotation and translation in the x-direction, the same y-direction, the same as the z-direction. so anybody in space has freedom of up to 6 degrees.

5. What are the Degrees of Freedom in Mathematics?

The level of freedom, in statistics, or any number of independent values ​​required to express the values ​​of all the variables of the system. The number of degrees of freedom is reduced by factors such as the need for a point to go in a certain direction.

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

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 + 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)

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

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.

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

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).