Simple Linear Regression Calculator

The simple linear regression model is expressed as:

$$Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i$$

Where \(\beta_0\) is the population intercept, \(\beta_1\) is the population slope, and \(\varepsilon_i\) is the random error term. The ordinary least squares estimates are:

$$b_1 = \frac{S_{XY}}{S_{XX}} = \frac{\sum(X_i - \bar{X})(Y_i - \bar{Y})}{\sum(X_i - \bar{X})^2}$$

$$b_0 = \bar{Y} - b_1 \bar{X}$$

The standard error of the slope measures the uncertainty in \(b_1\):

$$SE(b_1) = \sqrt{\frac{MSE}{S_{XX}}} = \sqrt{\frac{SSE/(n-2)}{\sum(X_i - \bar{X})^2}}$$

Where SSE (Sum of Squared Errors) is the total unexplained variation: \(SSE = S_{YY} - b_1 \cdot S_{XY}\), and MSE (Mean Squared Error) divides SSE by the degrees of freedom (n - 2). The t-statistic for testing \(H_0: \beta_1 = 0\) is:

$$t = \frac{b_1}{SE(b_1)}$$

This t-statistic follows a t-distribution with n - 2 degrees of freedom. A large absolute t-value (typically |t| > 2 for moderate sample sizes) provides evidence against the null hypothesis, suggesting the slope is significantly different from zero and X has a genuine linear effect on Y. The R² value quantifies model fit: \(R^2 = 1 - SSE/S_{YY}\), representing the fraction of Y's total variance explained by the regression.

The outputs of simple linear regression provide a complete picture of the relationship:

• Slope (b₁): The estimated change in Y per unit change in X. If b₁ = 1.85, then each one-unit increase in X is associated with a 1.85-unit increase in Y, on average.
• Intercept (b₀): The estimated Y when X = 0. May or may not be meaningful depending on whether X = 0 is within the data range.
• R²: Proportion of variance explained. Values closer to 1 indicate a better fit. A low R² does not necessarily mean the model is useless — even modest R² can be practically significant in noisy fields like social science.
• SE of Slope: Smaller values indicate more precise slope estimates. The SE decreases with more data points and when X values are more spread out.
• t-Statistic: Used to test \(H_0: \beta_1 = 0\). Compare |t| against critical t-values with n-2 degrees of freedom. For n=5 and α=0.05 (two-tailed), the critical value is approximately 3.182.

To make valid inferences, the standard regression assumptions must hold: linearity, independence, homoscedasticity, and normality of residuals. With only 5 data points, these assumptions are difficult to verify, so treat results as exploratory rather than definitive. For rigorous analysis, larger sample sizes are strongly recommended.