The concept of degrees of freedom (df) is fundamental and commonly used in statistical analysis. In this blog post, let us thoroughly understand this concept with examples.

Many a time, you might have encountered that the degrees of freedom for a particular test is, let's say (n-1) or (n-2) etc. In the blog post, let us understand this concept with examples.

**A) Without any restriction**

Suppose I give you three boxes as shown below. You are free to fill all these three boxes with any values of your choice. Hence, in this case, the degrees of freedom are 3. In other words, df=n here.

**B) With a restriction**

**you have to fill the third cell with 15**.

Hence, even though there were three cells, you were free to choose values for 2 cells. Hence, in this case, the degrees of freedom is 2. In other words,

**df=(n-1)**here.

**C) Degrees of freedom in contingency tables**

In contingency tables, degrees of freedom are = (row-1)*(column-1). Let us understand this using different examples.

**1) Degrees of freedom in ****2x2 contingency table**

Suppose you are given with a 2x2 table with row and column totals. You have to fill the values in four cells, but the row and column totals should be equal to the given values.

**2) Degrees of freedom in 2x3 contingency table**

**3) Degrees of freedom in 3x3 contingency table**

**D) Bessel's correction**

The formula for estimating sample variance is given below. Sample variance is the square of the sample standard deviation.

As you can see in the denominator, **(n-1)** is used instead of **n**.

What could be the reason? When we are estimating sample variance (or the sample standard deviation), we also need sample mean. For estimating the sample mean, **we have already used one of the degrees of freedom**.

Hence, now we are left with **(n-1)** degrees of freedom for estimating the sample variance (or the sample standard deviation). This is called Bessel's correction.

In summary, **degrees of freedom = (sample size) - (number of parameters to be estimated)**