What Is Degrees of Freedom and Why Does It Matter?

July 27, 2023
Degrees of freedom (DF) is the greatest possible number of rationally distinct values that can fluctuate in a data set. Degrees of freedom are determined by removing one from the total quantity of goods in the data sets batch. The initial notion of degrees of freedom was established at the beginning of the 1800s by the discoveries of mathematician and astronomer Carl Friedrich Gauss. Degrees of freedom are frequently mentioned in many kinds of assumption evaluation in research, such as chi-square. Degrees of freedom can be employed to define workplace conditions in that supervisor has to make a choice that affects the result of another aspect.

What are degrees of freedom?

Degrees of freedom is a scientific formula used mostly in research but also in physics and mechanics. The degrees of freedom in a statistical computation represent the number of possible values in a computation. The concept of DF covers the idea that the quantity of independent information available restricts the number of factors that may be estimated. The degrees of freedom are typically comparable to the number of samples minus the number of variables you must compute during an analysis. It is generally a complete number that is positive.

Degrees of freedom are determined by the amount of data available and the number of variables to be estimated. It displays how much independent information is used to estimate a parameter. In this sense, it is clear that a lot of information should go into parameter estimations to achieve more exact estimates and more effective hypothesis tests. The degrees of freedom can be estimated to assure the statistical reliability of t-tests, chi-square tests, and more sophisticated f-tests.

The background of degrees of freedom

Degrees of freedom were first identified in the early 1800s, interwoven in the writings of mathematician and astronomer Carl Friedrich Gauss. The current definition and application of the phrase were started by William Sealy Gosset, an English statistician, in his piece "The Probable Error of a Mean," released in Biometrika in 1908 under an alias to protect his privacy. Gosset did not use the word "degrees of freedom" in his publications. He did describe the notion across the development of what came to be known as "Student's T-distribution." The word did not become common until 1922. When he presented statistics and reports on his studies establishing chi-squares, English scientist and statistician, Ronald Fisher coined the phrase "degrees of freedom."

The concept of degrees of freedom

Degrees of freedom are the number of individual variables that may be evaluated in a statistical study and indicate the number of things that can be unpredictably chosen before limits must be imposed. Some establishing figures inside a data collection can be picked randomly. Suppose the data set is required to be equal to a specified total or mean, for instance. In that case, the number in the data set is restricted to assess the values of all other values in the data set, then fulfill the designated requirements. The number of variables that have the flexibility to fluctuate in computation is represented by degrees of freedom, which are terms related to mathematics in statistical analysis.

Evaluating degrees of freedom, among other things, may assist in confirming the reliability of chi-square test statistics, t-tests, and especially f-tests. These tests frequently contrast discovered data with predicted data if a specific assumption is accurate. Because the statistical degrees of freedom showing the number of variables in the last computation can fluctuate, they can add to the result's accuracy. Even though the number of sightings and parameters to be evaluated is determined by the scope of the sample, or the range of views and parameters, to be assessed, the degree of freedom in the computations is typically equivalent to the number of findings, less the number of parameters. This indicates that there are degrees of freedom open to higher sample sizes

The formula of degrees of freedom

The formula below is used to compute the degrees of freedom:

Df​ = N − 1


  • Df​ = degrees of freedom
  • N = sample size​

Some computations of degrees of freedom with numerous variables or connections utilize the formula

Df = N - P,

where P denotes the number of variables or relations. In a 2-sample t-test, for instance, N - 2 is employed since there are two variables to estimate.

Application of degrees of freedom

Degrees of freedom in statistics specify the form of the t-distribution used in t-tests to calculate the p-value. Distinct degrees of freedom produces distinct t-distributions based on the sample size. Comprehending the significance of a chi-square statistic and the reliability of the unproven theory involves determining degrees of freedom. Other than statistics, degrees of freedom have theoretical uses. Assume a corporation is making a purchasing decision for inputs for its production operations. Within this data set, the corporation has a pair of items, the quantity of raw materials to obtain, and the general price. The corporation can choose one of the two goods, but their decision will determine the fate of the other. It has one degree of freedom in this instance since it can only, without restriction, pick one of the two options. The corporation cannot identify the entire amount invested if it determines the number of raw materials. The firm's ability to obtain raw materials may be restricted if the total sum to spend is specified.

Chi-square test

The degree of freedom in this evaluation relies on the total number of groupings, and it is implemented to detect if there is an important relationship between two types of variables. Degrees of freedom in a chi-square test are calculated as (r - 1) x (c - 1), where r is the sum of rows in the table of contingencies and c is the sum of columns.

Chi-square tests are classified into two types: tests of independence and tests of association, such as "Is there a relationship between gender and SAT scores?"; and the goodness-of-fit test, which asks questions like "When a coin is tossed ten times will it come up heads five times and tails five times?" Degrees of freedom are used in these experiments to assess if the null theory can be discarded according to the overall quantity of variables and samples in the test. Obtaining the same or equivalent results from a research sample size of 400 or greater is more logical.


A T-test examines data collected from two comparable or dissimilar groups to assess the likelihood of a different outcome from what is typically achieved. The test's reliability is affected by several variables, such as the distribution patterns utilized and the variations impacting the gathered samples. The test is run based on the variables, and a T-value can be derived as a statistical inference of the likelihood of the typical consequence being driven at random.

The ultimate T-test determination might be reached using one of two methods:

·       A null hypothesis is one in which the difference between the means is 0, and the means are proved to be equal.

·       An alternative hypothesis suggests that the difference between the means is greater than zero. This contradicts the null hypothesis, suggesting that the information set is accurate and was not generated by chance.

Conversely, this T-test is legitimate and should be used when comparing the mean or average of two distinct groups or categories. When the number of comparisons to be conducted exceeds two, this method is not advised.

To run a t-test, compute the sample's t-value and contrast it to an essential value. The fundamental value will differ, and you can find the right value by utilizing a data set's t distribution with degrees of freedom. Groups with fewer degrees of freedom are more likely to contain high values. In contrast, groups with more degrees of freedom, like a minimum sample size of 30, will be considerably closer to a standard distribution curve. Reduced sample sizes equate to fewer degrees of freedom and wider t-distribution tails.

Linear regression

Calculating degrees of freedom in linear regression is a little more complex. However, it can be simplified. Every concept in a linear regression model is an approximated variable with one degree of freedom.

The error degrees of freedom in linear regression is the independent data units that can be used for predicting your coefficients. In regression, numerous error degrees of freedom is required for exact coefficient estimates and strong hypothesis tests, which equals many datasets for each model component.

The error degrees of freedom decline as more terms are added to the model. You have fewer data points from which to determine the coefficients. The accuracy of the estimations and the power of the tests are reduced due to this condition.


Degrees of freedom in an F-test relate to the independent samples used to determine a group's deviation. In particular, there are a pair of sets of degrees of freedom in an F-test: degrees of freedom for the numerator and degrees of freedom for the denominator. The number of independent observations needed to estimate the variance of the first group is represented by the numerator degrees of freedom (DFn). The number of independent samples used for estimating the variance of the second group or population is represented by the denominator degrees of freedom (DFd). The numerator's degrees of freedom are calculated as n1 - 1, and the denominator's degrees of freedom are calculated as n2 - 1, where n1 and n2 are the numbers of samples in the two categories (being analyzed) associated with the numerator and denominator, respectively.

The relevance of degrees of freedom in the real-world

Even though the degree of freedom is a theoretical concept commonly referenced in statistics, it is extremely useful in the real world. For instance, when recruiting labor to create output, company owners must consider two variables: work and output. Furthermore, the limitation is the connection between employees and production (the quantity of output employees can create).

In such a circumstance, entrepreneurs may either agree on the amount of output to be created, which determines the number of workers to be employed, or they may choose the number of staff members, which determines the number of goods produced. Therefore, the proprietors possess one degree of freedom regarding output and personnel.

Examples of degrees of freedom frequently asked questions

Is there always one degree of freedom?

The quantity of degrees of freedom is always the number of units in a given set minus one. It is always minus one because if parameters are applied to the data set, the final data item must be specified for every other aspect to comply with that result.

How are degrees of freedom computed?

Degrees of freedom are determined as the number of items in a group minus one when calculating the data collection average. This is because all things in that collection can be chosen randomly to the point only one is left; that one item has to comply with a particular mean.

What do degrees of freedom indicate?

Degrees of freedom indicates the number of units inside an array that may be chosen without limits while adhering to a specified law governing the entire collection. Assume a group of five objects with a mean value of 20. They signify the quantity of the (4) elements that may be randomly chosen before restrictions must be imposed. In the above instance, once the first four items are selected, you can no longer choose a data point at random since it's necessary to "force balance" to the specified average.


Degrees of freedom depict the quantity of information available concerning the number of attributes to be estimated. If you do not have enough data, your estimates will be inaccurate, and your statistical power will be poor. Various statistical analysis methods may need an indicator of the number of independent outcomes that can fluctuate within an assessment to fulfill constraint criteria. The degrees of freedom are the number of divisions in a representative population that can be picked arbitrarily before a particular number is selected.