A **Degrees of Freedom (DoF) Calculator** is a **statistical calculator** used to compute the number of **independent values** that can vary in an analysis without violating any **constraints**.

**DoF **are particularly important in **hypothesis testing**, **regression analysis**, and **chi-square tests**, among others.

The calculator simplifies the process of determining DoF, which can be complex depending on the specific **statistical test** being performed. By inputting relevant parameters such as **sample size**, **number of groups**, or **number of variables**, the calculator quickly computes the appropriate DoF value. This value is then used in conjunction with other **statistical measures** to assess the **significance of results** or to determine **critical values** in statistical tables.

## Degrees of Freedom Calculation Chart

Statistical Test/Situation | Degrees of Freedom Formula | Example Calculation |
---|---|---|

One-sample t-test | n – 1 | For 20 samples: DoF = 20 – 1 = 19 |

Two-sample t-test (equal variances) | (n₁ + n₂) – 2 | For groups of 15 and 20: DoF = (15 + 20) – 2 = 33 |

Paired t-test | n – 1 | For 25 pairs: DoF = 25 – 1 = 24 |

One-way ANOVA | (k – 1) and (N – k) | For 4 groups, total 60 samples: Between groups DoF = 4 – 1 = 3, Within groups DoF = 60 – 4 = 56 |

Chi-square test | (r – 1)(c – 1) | For 3×4 table: DoF = (3 – 1)(4 – 1) = 6 |

Simple linear regression | n – 2 | For 30 data points: DoF = 30 – 2 = 28 |

## Degrees of Freedom Formula

Let’s consider the formula for a **one-sample t-test**:

**DoF = n - 1**

Where **n** is the **sample size**.

Suppose we’re conducting a **one-sample t-test** with 25 observations. The degrees of freedom would be:

`DoF = 25 - 1 = 24`

**This means there are 24 independent values that can vary freely in this analysis.**

## How do you calculate degrees of freedom?

Calculating **degrees of freedom** involves identifying the number of **independent values** that can vary in a **statistical analysis**.

For a **chi-square test**, the formula is:

**DoF = (r - 1)(c - 1)**

Where **r** is the number of **rows** and **c** is the number of **columns** in the **contingency table**.

Imagine we’re analyzing the relationship between **gender** (male/female) and preference for three types of **music** (rock/pop/classical). Our **contingency table** has 2 rows (genders) and 3 columns (music types).

Calculating DoF:

r = 2(male, female)c = 3(rock, pop, classical)

`DoF = (2 - 1)(3 - 1) = 1 × 2 = 2`

This **chi-square test** has **2 degrees of freedom**. This value would be used to look up the **critical value** in a **chi-square distribution table** or to interpret the **p-value** in statistical software output.