Degrees of Freedom Calculator
Calculate degrees of freedom for various statistical tests and analyses
Statistical Test Parameters
Degrees of Freedom Results
Explanation
For a one-sample t-test with 10 observations, the degrees of freedom is 9.
Step-by-Step Calculation
1. Sample size (N) = 10
2. df = N - 1 = 10 - 1 = 9
Example Calculation
One-Sample t-Test
Sample size: N = 20
Formula: df = N - 1
Result: df = 20 - 1 = 19
Two-Sample t-Test
N₁ = 15, N₂ = 18
df = N₁ + N₂ - 2 = 15 + 18 - 2 = 31
Chi-Square Test
3 × 4 contingency table
df = (3-1) × (4-1) = 2 × 3 = 6
Common Test Types
One-Sample t-Test
Compare sample mean to population mean
df = N - 1
Two-Sample t-Test
Compare means of two groups
df = N₁ + N₂ - 2
ANOVA
Compare means of multiple groups
df = N - 1
Chi-Square
Test independence in contingency tables
df = (r-1)(c-1)
Key Concepts
Definition
Number of independent values that can vary in a calculation
Importance
Determines critical values and p-values in hypothesis testing
General Rule
Usually sample size minus number of parameters estimated
Understanding Degrees of Freedom
What are Degrees of Freedom?
Degrees of freedom (df) represent the number of independent pieces of information available to estimate a statistic. They indicate how many values in a calculation are free to vary after certain constraints have been applied. This concept is fundamental in statistical inference and hypothesis testing.
Why are They Important?
- •Determine the shape of sampling distributions
- •Affect critical values in hypothesis tests
- •Influence p-values and confidence intervals
- •Account for estimation uncertainty
Formula Summary
One-Sample Tests
df = N - 1
Where N is the sample size
Two-Sample Tests
df = N₁ + N₂ - 2
For equal variances
Regression Analysis
df = N - p - 1
Where p is number of predictors
General Principles
- •df = sample size - constraints
- •Constraints are estimated parameters
- •More constraints = fewer degrees of freedom
- •Larger df = more precise estimates
Common Applications
- •t-tests for mean comparisons
- •Chi-square tests for independence
- •F-tests in ANOVA
- •Regression model validation
Practical Tips
- •Higher df = more reliable results
- •Check df before interpreting p-values
- •Consider sample size in study design
- •df affects critical value selection