Computer 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):

Unit Converter ▲

Unit Converter ▼

From:	To:

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 regression line has the form y = mx + b, where m is the slope and b is the y-intercept.

2. How Does the Calculator Work?

The calculator uses the least squares method formulas:

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

Where:

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

Explanation: These formulas calculate the best-fit line that minimizes the sum of squared residuals between observed and predicted y values.

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, engineering, and social sciences.

4. Using the Calculator

Tips: Enter the summary statistics from your dataset (n, Σx, Σy, Σxy, Σx²). Ensure you have at least 2 data points (n ≥ 2) and that the x values aren't all identical.

5. Frequently Asked Questions (FAQ)

Q1: What does the slope (m) represent?
A: The slope indicates how much y changes for each unit change in x. A positive slope means y increases with x, negative means y decreases with x.

Q2: What does the intercept (b) represent?
A: The intercept is the predicted y value when x = 0. Interpretation depends on whether x = 0 is meaningful in your context.

Q3: How many data points do I need?
A: At least 2 points are required, but more points increase the reliability of your regression results.

Q4: What if I get "denominator is zero" error?
A: This occurs when all x values are identical, resulting in a vertical line (infinite slope). Check your data for this special case.

Q5: Can I use this for multiple regression?
A: No, this calculator is for simple linear regression (one x variable). Multiple regression requires matrix algebra.