Firstly let us introduce to you our Degrees of Freedom Calculator. This degrees of freedom calculator will assist you in calculating this critical variable for one- and two-sample t-tests, chi-square tests, and ANOVA. To discover out, read the following:

- What is a degree of freedom (definition of degrees of freedom);
- How to Calculate Degrees of Freedom; and
- The formula for degrees of freedom.

## What are degrees of freedom? Definition

Furthermore, degrees of freedom are associated with the maximum number of logically independent values in a data sample, with the freedom to fluctuate.

- Degrees of freedom relates to the maximum number of logically independent values in a data sample, with the freedom to fluctuate.
- Degrees of freedom are frequently mentioned in statistics concerning various types of hypothesis testing, such as chi-square.
- When attempting to understand the significance of a chi-square statistic and the validity of the null hypothesis, calculating degrees of freedom is critical.

That may sound very theoretical, but consider the following example:

Assume we have two numbers, x and y, and the mean of those two values, m. How many degrees of freedom do we have in our three-variable data set? The correct answer is 2. Why? Because the number of values that can change is two. The third variable is already decided if you pick the first two values. Look:

If x = 2 and y = 4, you can’t choose any mean; it’s already determined:

m = (x + y)/2

= (2 + 4)/2

= 3

When we assign 3 to x and 6 to m, the value of y is “automatically” established – it cannot be changed – because m = (x + y) / 2

- 6 = (3 + y) / 2
- 12 = 3 + y
- 12 – 3 = y
- y = 9

When two values are assigned, the third has no “freedom to alter,” hence there are two degrees of freedom in our example. Now that we understand what degrees of freedom are let’s look at calculating -df.

## How to find degrees of freedom – formulas

The degrees of freedom formula varies depending on the statistical test type being performed. However, the following are the equations for the most common ones:

**1-sample t-test**:

where: df – **Degrees of freedom**

**N** – denotes the total number of **subjects/values**.

**2-sample t-test (equal variance samples)**:

where:

N_1 denotes the number of values from the first sample; and

N_2 denotes the number of values from the second sample.

**Welch’s t-test (two-sample t-test with unequal variances)**:

In this scenario, we compute an estimate of the degrees of freedom as follows:

df \approx (\frac{\sigma_1}{N_1} + \frac{\sigma_2}{N_2})^2 / [\sigma_1^2 / (N_1^2 \cdot (N_1 - 1)) + \sigma_2^2 / (N_2^2 \cdot(N_2 - 1))],where \sigma – Variation

**ANOVA**:- Differential degrees of freedom between groups:

Where k is the **number of groups of cells**.

**Groups’ degrees of freedom**:

The **total number** of degrees of freedom:

**Chi – square test test**:

If you’re looking for a quick way to find df, utilize our degrees of freedom calculator. It incorporates all of the preceding formulae. Check out our **chi-square calculator**!

## Degrees of freedom calculator

The df calculator is used as follows:

- First, select the statistical test you’ll be employing.
- Fill in the variables displayed in the rows below, such as the sample size.
- The answer is in the last box of the df calculator.

## Degrees Of Freedom – One Sample

The difference between the sample average and the null hypothesis value is statistically significant when using a **one-sample t-test**. Let’s return to our nasty example from before. When we have a sample and estimate the mean, we know that we have n – 1 degrees of freedom, where n is the sample size. As a result, the degree of freedom for a one-sample t-test is n – 1.

The DF specifies the form of the t-distribution used by your t-test to get the p-value. The graph below depicts the t-distribution for various degrees of freedom. Because the degrees of freedom are closely tied to sample size, the influence of sample size may be seen. The t-distribution has thicker tails as the DF drops. This feature allows for the higher level of uncertainty that comes with smaller sample sizes.

## Degrees of Freedom – Two Samples

The t-test yields two results: the t-value and the degrees of freedom. The t-value is a ratio of the difference in mean between the two sample sets and the variance within the sample sets. While the numerator value (the difference between the means of the two sample sets) is simple to compute, the denominator (the variance within the sample sets) can become complex depending on the data values involved. The ratio’s denominator is a measure of dispersion or variability. Higher t-values, also known as t-scores, significantly differ between the two sample sets. Conversely, the lower the t-value, the greater the similarity between the two sample sets.

- A high t-score implies that the groups are dissimilar.
- A low t-score shows that the groups are comparable.

Degrees of freedom are the values in research that have the flexibility to fluctuate and are critical for determining the significance and validity of the null hypothesis. The quantity of data records available in the sample set generally determines how these values are computed.

## Degrees Of Freedom – Chi-Square

Experiments validate predictions. These predictions are frequently numerical, which means that when scientists collect data, they anticipate the numbers to break down in a specific way. Unfortunately, real-world data seldom matches the exact predictions made by scientists. Thus scientists require a test to determine if the gap between observed and predicted numbers is due to random chance or some unforeseen aspect that will lead the scientist to change the underlying theory. A chi-square test is a statistical method that scientists employ to do this.

**What Kind of Information we Need**

A chi-square test requires categorical data. The number of individuals who replied “yes” vs the number of people who said “no” (two categories) is an example of categorical data, as is the number of frogs in a population that are green, yellow, or grey (three categories). A chi-square test cannot be used on continuous data, such as that obtained from a poll asking respondents how tall they are. A survey of this nature would provide a wide range of heights. However, if we separate heights into groups such as “under 6 feet tall” and “6 feet tall and more,” the data might be subjected to a chi-square test.

**The Chi-Square Statistic and Its Interpretation**

The chi-square statistic indicates how far your observed values differed from your projected values—the more significant the difference, the greater the number. You can tell whether your chi-square value is too high or too low to support your prediction by looking at a chi-square distribution table and checking if it is below a particular critical number. This table correlates chi-square values with probabilities known as p-values. The table tells you explicitly if the variations between your observed and predicted numbers are purely due to random chance or whether another factor is present. For example, if the p-value for a goodness-of-fit test is 0.05 or below, your prediction must be rejected.

## Degrees Of Freedom in ANOVA

**ANOVA** may be used to examine if the means of three or more groups differ. F-tests are used in ANOVA to assess the equality of means statistically. In this post, I’ll use a one-way ANOVA example to demonstrate how ANOVA and F-tests operate.

**F-tests** are called after the F statistic, named after Sir Ronald Fisher. The F-statistic is nothing more than a ratio of two variances. Variances measure dispersion, or how far apart the data are from the mean. Higher values show more excellent distribution.

The variance is defined as the square of the standard deviation. Because they are in the same units as the data rather than squared units, standard deviations are more accessible for us humans to grasp than variances. However, variances are included in the computations in many analyses.

The ratio of mean squares is used to calculate F-statistics. The word “mean squares” may sound perplexing. Still, it merely refers to an estimate of population variance that considers the degrees of freedom (DF) utilized to compute that estimate.

## Why Do Critical Values Decrease While DF Increase?

Degrees of freedom is proportional to sample size (n-1). As the df grows, so does the sample size; the t-distribution graph will have skinnier tails, moving the crucial value closer to the mean.

## Df chart

DF | A = 0.1 | 0.05 | 0.025 | 0.01 | 0.005 | 0.001 | 0.0005 |

∞ | t_{a} = 1.282 | 1.645 | 1.960 | 2.326 | 2.576 | 3.091 | 3.291 |

1 | 3.078 | 6.314 | 12.706 | 31.821 | 63.656 | 318.289 | 636.578 |

2 | 1.886 | 2.920 | 4.303 | 6.965 | 9.925 | 22.328 | 31.600 |

3 | 1.638 | 2.353 | 3.182 | 4.541 | 5.841 | 10.214 | 12.924 |

4 | 1.533 | 2.132 | 2.776 | 3.747 | 4.604 | 7.173 | 8.610 |

5 | 1.476 | 2.015 | 2.571 | 3.365 | 4.032 | 5.894 | 6.869 |

6 | 1.440 | 1.943 | 2.447 | 3.143 | 3.707 | 5.208 | 5.959 |

7 | 1.415 | 1.895 | 2.365 | 2.998 | 3.499 | 4.785 | 5.408 |

8 | 1.397 | 1.860 | 2.306 | 2.896 | 3.355 | 4.501 | 5.041 |

9 | 1.383 | 1.833 | 2.262 | 2.821 | 3.250 | 4.297 | 4.781 |

10 | 1.372 | 1.812 | 2.228 | 2.764 | 3.169 | 4.144 | 4.587 |

11 | 1.363 | 1.796 | 2.201 | 2.718 | 3.106 | 4.025 | 4.437 |

12 | 1.356 | 1.782 | 2.179 | 2.681 | 3.055 | 3.930 | 4.318 |

13 | 1.350 | 1.771 | 2.160 | 2.650 | 3.012 | 3.852 | 4.221 |

14 | 1.345 | 1.761 | 2.145 | 2.624 | 2.977 | 3.787 | 4.140 |

15 | 1.341 | 1.753 | 2.131 | 2.602 | 2.947 | 3.733 | 4.073 |

16 | 1.337 | 1.746 | 2.120 | 2.583 | 2.921 | 3.686 | 4.015 |

17 | 1.333 | 1.740 | 2.110 | 2.567 | 2.898 | 3.646 | 3.965 |

18 | 1.330 | 1.734 | 2.101 | 2.552 | 2.878 | 3.610 | 3.922 |

19 | 1.328 | 1.729 | 2.093 | 2.539 | 2.861 | 3.579 | 3.883 |

20 | 1.325 | 1.725 | 2.086 | 2.528 | 2.845 | 3.552 | 3.850 |

21 | 1.323 | 1.721 | 2.080 | 2.518 | 2.831 | 3.527 | 3.819 |

22 | 1.321 | 1.717 | 2.074 | 2.508 | 2.819 | 3.505 | 3.792 |

23 | 1.319 | 1.714 | 2.069 | 2.500 | 2.807 | 3.485 | 3.768 |

24 | 1.318 | 1.711 | 2.064 | 2.492 | 2.797 | 3.467 | 3.745 |

25 | 1.316 | 1.708 | 2.060 | 2.485 | 2.787 | 3.450 | 3.725 |

26 | 1.315 | 1.706 | 2.056 | 2.479 | 2.779 | 3.435 | 3.707 |

27 | 1.314 | 1.703 | 2.052 | 2.473 | 2.771 | 3.421 | 3.689 |

28 | 1.313 | 1.701 | 2.048 | 2.467 | 2.763 | 3.408 | 3.674 |

29 | 1.311 | 1.699 | 2.045 | 2.462 | 2.756 | 3.396 | 3.660 |

30 | 1.310 | 1.697 | 2.042 | 2.457 | 2.750 | 3.385 | 3.646 |

60 | 1.296 | 1.671 | 2.000 | 2.390 | 2.660 | 3.232 | 3.460 |

120 | 1.289 | 1.658 | 1.980 | 2.358 | 2.617 | 3.160 | 3.373 |

1000 | 1.282 | 1.646 | 1.962 | 2.330 | 2.581 | 3.098 | 3.300 |

**T-Distribution Table (One Tail)**

DF | A = 0.2 | 0.10 | 0.05 | 0.02 | 0.01 | 0.002 | 0.001 |

∞ | t_{a} = 1.282 | 1.645 | 1.960 | 2.326 | 2.576 | 3.091 | 3.291 |

1 | 3.078 | 6.314 | 12.706 | 31.821 | 63.656 | 318.289 | 636.578 |

2 | 1.886 | 2.920 | 4.303 | 6.965 | 9.925 | 22.328 | 31.600 |

3 | 1.638 | 2.353 | 3.182 | 4.541 | 5.841 | 10.214 | 12.924 |

4 | 1.533 | 2.132 | 2.776 | 3.747 | 4.604 | 7.173 | 8.610 |

5 | 1.476 | 2.015 | 2.571 | 3.365 | 4.032 | 5.894 | 6.869 |

6 | 1.440 | 1.943 | 2.447 | 3.143 | 3.707 | 5.208 | 5.959 |

7 | 1.415 | 1.895 | 2.365 | 2.998 | 3.499 | 4.785 | 5.408 |

8 | 1.397 | 1.860 | 2.306 | 2.896 | 3.355 | 4.501 | 5.041 |

9 | 1.383 | 1.833 | 2.262 | 2.821 | 3.250 | 4.297 | 4.781 |

10 | 1.372 | 1.812 | 2.228 | 2.764 | 3.169 | 4.144 | 4.587 |

11 | 1.363 | 1.796 | 2.201 | 2.718 | 3.106 | 4.025 | 4.437 |

12 | 1.356 | 1.782 | 2.179 | 2.681 | 3.055 | 3.930 | 4.318 |

13 | 1.350 | 1.771 | 2.160 | 2.650 | 3.012 | 3.852 | 4.221 |

14 | 1.345 | 1.761 | 2.145 | 2.624 | 2.977 | 3.787 | 4.140 |

15 | 1.341 | 1.753 | 2.131 | 2.602 | 2.947 | 3.733 | 4.073 |

16 | 1.337 | 1.746 | 2.120 | 2.583 | 2.921 | 3.686 | 4.015 |

17 | 1.333 | 1.740 | 2.110 | 2.567 | 2.898 | 3.646 | 3.965 |

18 | 1.330 | 1.734 | 2.101 | 2.552 | 2.878 | 3.610 | 3.922 |

19 | 1.328 | 1.729 | 2.093 | 2.539 | 2.861 | 3.579 | 3.883 |

20 | 1.325 | 1.725 | 2.086 | 2.528 | 2.845 | 3.552 | 3.850 |

21 | 1.323 | 1.721 | 2.080 | 2.518 | 2.831 | 3.527 | 3.819 |

22 | 1.321 | 1.717 | 2.074 | 2.508 | 2.819 | 3.505 | 3.792 |

23 | 1.319 | 1.714 | 2.069 | 2.500 | 2.807 | 3.485 | 3.768 |

24 | 1.318 | 1.711 | 2.064 | 2.492 | 2.797 | 3.467 | 3.745 |

25 | 1.316 | 1.708 | 2.060 | 2.485 | 2.787 | 3.450 | 3.725 |

26 | 1.315 | 1.706 | 2.056 | 2.479 | 2.779 | 3.435 | 3.707 |

27 | 1.314 | 1.703 | 2.052 | 2.473 | 2.771 | 3.421 | 3.689 |

28 | 1.313 | 1.701 | 2.048 | 2.467 | 2.763 | 3.408 | 3.674 |

29 | 1.311 | 1.699 | 2.045 | 2.462 | 2.756 | 3.396 | 3.660 |

30 | 1.310 | 1.697 | 2.042 | 2.457 | 2.750 | 3.385 | 3.646 |

60 | 1.296 | 1.671 | 2.000 | 2.390 | 2.660 | 3.232 | 3.460 |

120 | 1.289 | 1.658 | 1.980 | 2.358 | 2.617 | 3.160 | 3.373 |

8 | 1.282 | 1.645 | 1.960 | 2.326 | 2.576 | 3.091 | 3.291 |

**Two Tails T Distribution Table**

## Frequently Asked Questions

**How to calculate degrees of freedom for t-test?**

To compute the degrees of freedom for a one-sample t-test:

1. First, determine the size of your sample (N).

2. Subtract one.

3. The amount of degrees of freedom is the outcome.

**How to find degrees of freedom chi-square**

Use the following formula to determine degrees of freedom for the chi-square test:**df = (rows – 1) * (columns – 1)**,

that is:

1. Subtract one from the number of rows in the chi-square table.

2. Subtract one from the number of columns.

3. Multiply the result of step 1 by the result of step 2.

**How to calculate degrees of freedom for two-sample t-test?**

**df = N1 + N2 – 2, which means:**

1. Calculate the dimensions of your two samples.

2. Add them all up.

3. Add 2 to the previous step’s result.

**Can degrees of freedom be 0?**

Yes, theoretically, degrees of freedom can equal zero. It would imply that there is only one piece of data with no “freedom” to alter and no unknown variables. However, 0 degrees of freedom should not be used when running statistical tests.