Calculate Linear Regression Calculator

Linear Regression Formula:

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

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.

2. How Does the Calculator Work?

The calculator uses the linear regression formulas:

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

Where:

\( m \) — Slope of the regression line
\( b \) — Y-intercept of the regression line
\( n \) — Number of data points
\( x \) — Independent variable values
\( y \) — Dependent variable values

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

3. Importance of Linear Regression

Details: Linear regression is fundamental in statistics for understanding relationships between variables, making predictions, and testing hypotheses.

4. Using the Calculator

Tips: Enter comma-separated values for x and y variables. Both lists must be the same length. The calculator will compute the slope (m) and y-intercept (b) of the regression line.

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 as x increases, while a negative slope means y decreases as x increases.

Q2: What does the intercept (b) represent?
A: The intercept is the predicted value of y when x equals zero. It's where the regression line crosses the y-axis.

Q3: How many data points do I need?
A: While you can calculate regression with just two points, more points provide a more reliable estimate of the true relationship.

Q4: What assumptions does linear regression make?
A: Key assumptions include linearity, independence, homoscedasticity (constant variance), and normality of residuals.

Q5: How do I interpret R-squared?
A: R-squared (not shown here) measures the proportion of variance in y explained by x. Values range from 0 to 1, with higher values indicating better fit.