Explanation
In simple linear regression, the normality assumption specifically applies to the regression residuals (also called error terms), not to the dependent or independent variables themselves.
Key Points:
- Normality of Residuals: The classical linear regression model assumes that the error terms (ε) are normally distributed with mean zero and constant variance σ².
- Why Residuals, Not Variables:
- The dependent variable (Y) can have any distribution; it's the conditional distribution of Y given X that matters
- The independent variable (X) can be fixed or random; no normality assumption is required for X
- The residuals represent the unexplained variation after accounting for the linear relationship
- Importance: This assumption is crucial for:
- Valid hypothesis testing (t-tests, F-tests)
- Constructing confidence intervals
- Making predictions with proper uncertainty quantification
- Central Limit Theorem: Even if residuals aren't perfectly normal, with large sample sizes, inference is often robust due to the Central Limit Theorem.
Mathematical Representation:
The regression model: Y = β₀ + β₁X + ε
Where ε ~ N(0, σ²)
This means the residuals should be approximately normally distributed around zero.
