Slope Formula

m = Σ[(x - x̄)(y - ȳ)] / Σ(x - x̄)²

Or equivalently: m = [n(Σxy) - (Σx)(Σy)] / [n(Σx²) - (Σx)²]

Y-Intercept Formula

b = ȳ - m(x̄)

Where x̄ and ȳ are the means of x and y. The regression line always passes through the point (x̄, ȳ).

Example Calculation

Data: (1, 2), (2, 4), (3, 5), (4, 7)

• Mean: x̄ = 2.5, ȳ = 4.5
• Slope: m = 1.6
• Intercept: b = 0.5

Regression line: y = 1.6x + 0.5

R² (Coefficient of Determination)

R² tells you what percentage of variation in y is explained by x. Values range from 0 to 1.

• R² = 1.0: Perfect fit (all points on the line)
• R² = 0.7-0.9: Strong relationship
• R² = 0.4-0.7: Moderate relationship
• R² < 0.4: Weak relationship

Residuals

Residuals are the differences between actual y-values and predicted y-values. A good regression has randomly scattered residuals with no pattern. Patterns in residuals suggest the linear model may not be appropriate.

Sales Forecasting

Predict future sales based on historical data, advertising spend, or seasonal trends.

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Estimate home prices based on square footage, bedrooms, location, and other features.

Medical Research

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Climate Studies

Analyze temperature trends over time and predict future climate patterns.