Definition:

Degrees of freedom (df) is a measure of the number of independent variables that are free to vary in a sample, given the constraints imposed by the sample size and any other statistical assumptions.

Formula:

Degrees of freedom (df) = n – r

where:

• n is the sample size
• r is the number of parameters estimated from the sample (e.g., the number of intercepts and slopes in a regression model)

Explanation:

• Degrees of freedom (df) are used to calculate the degrees of freedom for a t-statistic, chi-square statistic, or F-statistic.
• The number of degrees of freedom is equal to the sample size minus the number of parameters estimated from the sample.
• For example, if a sample of 20 observations is used to estimate a mean and a standard deviation, then the degrees of freedom for the t-statistic would be 18.
• Degrees of freedom are important in statistical inference because they determine the critical values for probability distributions.

Examples:

• T-statistic: The degrees of freedom for a t-statistic are equal to the sample size minus 1.
• Chi-square statistic: The degrees of freedom for a chi-square statistic are equal to the number of categories minus 1.
• F-statistic: The degrees of freedom for an F-statistic are equal to the number of groups minus 1.

Additional Notes:

• Degrees of freedom can be positive or negative.
• Negative degrees of freedom are not valid.
• The degrees of freedom are used to calculate the p-value.
• The degrees of freedom are a key concept in statistics and are used in a wide variety of statistical tests.