Linear Regression Calculator

Calculate simple linear regression equation y = mx + b from data points. Find slope, intercept, R² goodness of fit, and predicted values. Identify trends in your data.

📊 Statistics📐 y = mx + b (m = r × sy/sx)🔢 Math
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Formula & Reference

VariableSymbolFormulaUnits
Linear Regression Calculator—y = mx + b (m = r × sy/sx)same as data units

Step-by-Step Examples

Example 1
Study Hours vs Test Score

x (hours): 1,2,3,4,5. y (score): 50,60,65,70,80.

  • mx = 3, my = 65
  • m = Σ(xi-mx)(yi-my)/Σ(xi-mx)² = 20/10 = 7
  • b = 65 - 7×3 = 44
  • Equation: y = 7x + 44
  • Each extra hour predicts 7 more points
✓ y = 7x + 44 (R² ≈ 0.94)
Example 2
House Size vs Price

x (sq ft): 1000,1200,1500,1800,2000. y ($k): 150,180,210,250,280.

  • Slope m ≈ 0.12 ($k per sq ft = $120/sq ft)
  • b ≈ 3.6 ($3,600 base)
  • Predict 1600 sq ft: y = 0.12×1600+3.6 = 195.6 ($196k)
  • R² ≈ 0.99: very strong linear fit
✓ y = 0.12x + 3.6, predict 1600 sqft → $196k
Example 3
Prediction from Regression

Same equation y=7x+44. Predict score for 8 hours studying.

  • y = 7(8) + 44 = 56 + 44 = 100
  • Predicted score = 100
  • But 8 hours is outside training data range (1-5 hrs)
  • Extrapolation beyond data range is risky!
✓ Predicted y(8) = 100 (be cautious: extrapolation)

Real-World Applications

💼
Business Forecasting
Sales trends, demand forecasting, cost estimation all use linear regression.
🏠
Real Estate
Price-per-sqft analysis, market trend fitting — linear regression is the foundation.
📊
Finance
Stock return vs market return (beta in CAPM), bond yield curves, option pricing.
🧪
Scientific Research
Calibration curves in chemistry, dose-response in medicine, environmental trend analysis.

Common Mistakes to Avoid

⚠️
Extrapolating beyond your data range

Predicting y for x values far outside your training data is unreliable. The relationship may be nonlinear outside the observed range.

⚠️
Ignoring residual plots

R² alone doesn't confirm a linear relationship is appropriate. Always check a residual plot for patterns — systematic curves indicate nonlinearity.

⚠️
Confusing regression with causation

Regression fits a line; it doesn't prove x causes y. Many strong regressions are spurious (ice cream sales predicting drownings — both caused by summer heat).

Frequently Asked Questions

What is simple linear regression? ▾
Models the relationship between two variables as y = mx + b. m is the slope (change in y per unit change in x), b is the y-intercept. Minimizes the sum of squared residuals.
What is R²? ▾
Proportion of variance in y explained by x. R²=0.80 means 80% of the variation in y is explained by x. The remaining 20% is unexplained (residual) variation.
What is the difference between regression and correlation? ▾
Correlation (r): symmetric, just measures relationship strength. Regression: directional (predicts y from x), gives the equation, can make predictions. R² = r² for simple regression.
What are the assumptions of linear regression? ▾
1) Linear relationship, 2) Independence of observations, 3) Homoscedasticity (constant variance of residuals), 4) Normality of residuals. Check with residual plots.
When should I use multiple regression? ▾
When you have more than one predictor variable. y = b0 + b1x1 + b2x2 + ... Accounts for multiple factors simultaneously. Requires matrix algebra or statistical software.

Related Math Calculators

📊 Correlation r📊 P-Value📊 Standard Error📊 Statistics📊 Variance

Formula Explorer connections

Interpretation: This formula summarizes data, models uncertainty or supports inference about a population or random process. Assumption: The sampling design and distribution assumptions must match the data. Independence, sample size, outliers and measurement quality can materially affect interpretation.

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