Degrees of Freedom

Reviewed by Sweta | Updated on Sep 14, 2021



Degrees of freedom is a statistical concept referring to the number of variable factors which are independent in a system, whose value could vary in any direction depending on the manner the movement occurs.

Technical analysis of the independent variables will specify the amount of freedom a given variable has to vary in different directions in a data sample.


An example to understand the behaviour of variables in a data sample is as below:

In a data sample consisting of five positive integers, we consider the value to be any positive number. The value of each integer does not bear any relation to another integer in the sample. From a theoretical perspective, the data sample will have five degrees of freedom.

In the sample, the four numbers are 3, 4, 5 and 8 and accordingly, the average of the entire sample data will be 6. A logical calculation of the sample indicates that the fifth number in the series is 10. Statistically, the number cannot vary from beyond five degrees. Hence, the number is 10.

Accordingly, in the sample, the degrees of freedom is 4.

Statistically, the technical formula to arrive at degrees of freedom takes into account, the size of the data sample as reduced by 1:

*Df =N−1 * where: Df = degrees of freedom ​N=sample size


Degrees of freedom is a statistical tool and hence finds its use in different types of hypothesis testing, such as Chi-Square tests. A person analysing data of different sample sizes will use the calculation of degrees of freedom while trying to understand the Chi-Square statistic and also the importance and validity of the null hypothesis.

Among the Chi-Square tests, there are of two types: 1. The independence test in which testing determines the relationship between variables. A testing example is "Is there a relationship between age and CET scores?"

  1. The goodness-of-fit test seeks to ask something like "If a coin is tossed 20 times, will head of the coin come up ten times and tails ten times?"


The history of the concept of degrees of freedom dates back to the early 1800s, intermingled as part of the works of Mr Carl Friedrich Gauss, who was a great mathematician and astronomer.

The current understanding and usage of the term 'Degrees of Freedom' had been given by Professor William Sealy Gosset, who was an English statistician. The professor wrote in his article "The Probable Error of a Mean," which got published in Biometrika in the year 1908 under a pen name to preserve his anonymity.

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