Linear Regression Calculator

What is Linear Regression?

Linear regression is a statistical method used to model the relationship between a dependent variable (Y) and one independent variable (X) by fitting a linear equation to observed data. The goal is to find the line that best fits the data points.

The Regression Equation

y = mx + b

Where:
m = slope (rate of change)
b = y-intercept (value of y when x = 0)

Least Squares Formulas

The slope and y-intercept are calculated using the least squares method:

Slope (m) = (n * sum(xy) - sum(x) * sum(y)) / (n * sum(x^2) - (sum(x))^2)

Y-intercept (b) = (sum(y) - m * sum(x)) / n

Or equivalently:
b = y_mean - m * x_mean

How to Calculate Linear Regression

1. Calculate sums: Find sum(x), sum(y), sum(xy), sum(x^2)
2. Calculate slope (m): Use the slope formula
3. Calculate y-intercept (b): Use the intercept formula
4. Write equation: y = mx + b
5. Make predictions: Substitute x values to predict y

Example Calculation

For data points: (1, 2), (2, 4), (3, 5), (4, 4), (5, 5)

n = 5
sum(x) = 15, sum(y) = 20
sum(xy) = 66, sum(x^2) = 55
x_mean = 3, y_mean = 4

Slope (m) = (5*66 - 15*20) / (5*55 - 225)
         = (330 - 300) / (275 - 225)
         = 30 / 50 = 0.6

Y-intercept (b) = 4 - 0.6*3 = 4 - 1.8 = 2.2

Equation: y = 0.6x + 2.2

Prediction: When x = 6
y = 0.6(6) + 2.2 = 3.6 + 2.2 = 5.8

Interpreting the Results

Slope (m)

• Positive slope: Y increases as X increases
• Negative slope: Y decreases as X increases
• Slope = 0: No relationship between X and Y
• The slope indicates how much Y changes for each unit increase in X

Y-Intercept (b)

• The value of Y when X equals zero
• May or may not have practical meaning depending on context

R-Squared (Coefficient of Determination)

R-squared measures how well the regression line fits the data. Values range from 0 to 1, where 1 indicates a perfect fit.

Applications of Linear Regression

Business

Sales forecasting, demand prediction, cost analysis.

Science

Analyzing experimental data, predicting outcomes.

Economics

Modeling economic relationships, trend analysis.

Engineering

Calibration curves, performance prediction.