Linear Regression Calculator Online – Free | 8gwifi.org

Linear Regression Calculator

Calculate regression equations, R², correlation, and make predictions with interactive visualizations

Data Input

Enter your data: Provide X and Y values as comma-separated pairs, one per line. Example: 1,2 or use the sample data button.

Results

Enter data and click Calculate to see results

Understanding Linear Regression

Linear regression is a statistical method for modeling the relationship between a dependent variable (Y) and an independent variable (X). It finds the best-fitting straight line through the data points.

The Regression Equation

The linear regression line is expressed as:

y = a + bx

Where:

  • y = predicted value (dependent variable)
  • x = independent variable (predictor)
  • b = slope (change in y for each unit change in x)
  • a = y-intercept (value of y when x = 0)

Key Calculations

Slope (b): b = Σ[(xᵢ - x̄)(yᵢ - ȳ)] / Σ(xᵢ - x̄)²
Intercept (a): a = ȳ - b × x̄
R² (Coefficient of Determination): R² = 1 - (SS_res / SS_tot)

Understanding R² (R-Squared)

R² measures how well the regression line fits the data:

R² Value Interpretation Fit Quality
0.90 - 1.00 90-100% of variance explained Excellent
0.70 - 0.89 70-89% of variance explained Good
0.50 - 0.69 50-69% of variance explained Moderate
0.00 - 0.49 Less than 50% explained Weak

Correlation Coefficient (r)

The correlation coefficient measures the strength and direction of the linear relationship:

  • r = +1: Perfect positive correlation
  • r = 0: No linear correlation
  • r = -1: Perfect negative correlation
  • |r| > 0.7: Strong correlation
  • 0.3 < |r| < 0.7: Moderate correlation
  • |r| < 0.3: Weak correlation

Standard Error of Estimate (SEE)

The standard error measures the typical distance between observed and predicted values:

SEE = √[Σ(yᵢ - ŷᵢ)² / (n - 2)]

A smaller SEE indicates better predictions.

Assumptions of Linear Regression

  • Linearity: The relationship between X and Y is linear
  • Independence: Observations are independent of each other
  • Homoscedasticity: Residuals have constant variance
  • Normality: Residuals are normally distributed
  • No outliers: Extreme values can disproportionately affect the line

Real-World Applications

  • Business: Sales forecasting, price optimization, demand prediction
  • Economics: GDP analysis, inflation prediction, market trends
  • Healthcare: Drug dosage, disease progression, risk factors
  • Science: Calibration curves, experimental relationships
  • Education: Grade prediction, study time vs. performance
  • Real Estate: Property valuation based on features

Interpreting Results

Slope Interpretation: "For every 1-unit increase in X, Y increases/decreases by b units."

Intercept Interpretation: "When X = 0, the predicted value of Y is a."

R² Interpretation: "The model explains R²×100% of the variance in Y."

Tip: Always visualize your data with a scatter plot before performing regression. Linear regression assumes a linear relationship - if your data shows a curved pattern, consider polynomial regression or data transformation.

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Linear Regression Calculator: FAQ

What does R² tell me?

R² is the proportion of variance in Y explained by X using the fitted linear model. Higher values indicate better fit, but check residuals for assumptions.

Are there key assumptions?

Linearity, independence, homoscedasticity (constant variance), and approximately normal residuals. Inspect residual plots for violations.

How do I make predictions?

Enter an X value; the model predicts Y = intercept + slope × X. Consider prediction intervals for uncertainty around point estimates.

What if the relationship isn’t linear?

Try transformations (e.g., log) or polynomial terms, or consider non‑linear or robust methods if residual patterns persist.