Find Regression Coefficient Calculator

Linear Regression Slope Formula:

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

Sample Size (n):

points

Sum of X Values (Σx):

Sum of Y Values (Σy):

Sum of XY Products (Σxy):

Sum of X² Values (Σx²):

Regression Coefficient (b):

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1. What is the Regression Coefficient?

The regression coefficient (b) represents the slope of the linear regression line, indicating the rate of change in the dependent variable (y) for each unit change in the independent variable (x). It quantifies the relationship between variables in linear regression analysis.

2. How Does the Calculator Work?

The calculator uses the linear regression slope formula:

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

Where:

\( b \) — Regression coefficient (slope)
\( n \) — Sample size
\( \sum x \) — Sum of all x values
\( \sum y \) — Sum of all y values
\( \sum xy \) — Sum of x*y products
\( \sum x^2 \) — Sum of squared x values

Explanation: This formula calculates the best-fit slope that minimizes the sum of squared residuals between observed and predicted values.

3. Importance of Regression Coefficient

Details: The regression coefficient is fundamental in statistical analysis for understanding relationships between variables, making predictions, and testing hypotheses about variable relationships.

4. Using the Calculator

Tips: Enter the sample size and all required sums. Ensure the denominator is not zero (which occurs when all x values are identical). The calculator provides the slope of the regression line.

5. Frequently Asked Questions (FAQ)

Q1: What does a positive regression coefficient indicate?
A: A positive coefficient indicates a positive relationship - as x increases, y tends to increase.

Q2: What does a negative regression coefficient indicate?
A: A negative coefficient indicates a negative relationship - as x increases, y tends to decrease.

Q3: How is this different from correlation coefficient?
A: Correlation measures strength of relationship, while regression coefficient measures the rate of change and allows for prediction.

Q4: When is the regression coefficient undefined?
A: When the denominator is zero, which happens when all x values are identical (zero variance in x).

Q5: Can this be used for multiple regression?
A: No, this formula is specifically for simple linear regression with one independent variable.