Regression Output Calculator

Linear Regression Formula:

\[ m = \frac{n \times \Sigma(xy) - \Sigma x \times \Sigma y}{n \times \Sigma(x^2) - (\Sigma x)^2} \] \[ b = \frac{\Sigma y - m \times \Sigma x}{n} \]

Slope (m):

Intercept (b):

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1. What is Linear Regression?

Linear regression is a statistical method that models the relationship between a dependent variable (y) and one or more independent variables (x) by fitting a linear equation to observed data. The calculator provides the slope (m) and intercept (b) of the best-fit line.

2. How Does the Calculator Work?

The calculator uses the ordinary least squares method:

\[ m = \frac{n \times \Sigma(xy) - \Sigma x \times \Sigma y}{n \times \Sigma(x^2) - (\Sigma x)^2} \] \[ b = \frac{\Sigma y - m \times \Sigma x}{n} \]

Where:

\( m \) — Slope of the regression line (unitless)
\( b \) — Y-intercept of the regression line (in units of y)
\( n \) — Number of data points
\( \Sigma xy \) — Sum of x*y products
\( \Sigma x \) — Sum of x values
\( \Sigma y \) — Sum of y values
\( \Sigma x^2 \) — Sum of squared x values

Explanation: The slope indicates how much y changes for each unit change in x, while the intercept represents the predicted y value when x is zero.

3. Importance of Regression Analysis

Details: Regression analysis is fundamental for understanding relationships between variables, making predictions, and testing hypotheses in fields like economics, biology, and social sciences.

4. Using the Calculator

Tips: Enter the sums of your data points (Σxy, Σx, Σy, Σx²) and the number of observations (n ≥ 2). The calculator will compute the slope and intercept of the best-fit line.

5. Frequently Asked Questions (FAQ)

Q1: What does the slope represent?
A: The slope indicates the rate of change in the dependent variable (y) per unit change in the independent variable (x).

Q2: When is linear regression appropriate?
A: When the relationship between variables appears linear, residuals are normally distributed, and homoscedasticity is present.

Q3: What's the minimum number of data points needed?
A: At least 2 points are required, but more points increase the reliability of the estimates.

Q4: How do I interpret the intercept?
A: The intercept represents the predicted y value when x=0, but may not always have practical meaning if x=0 is outside the observed range.

Q5: What if my denominator is zero?
A: This occurs when all x values are identical, resulting in a vertical line (undefined slope).