This calculator performs linear regression by calculating the slope, intercept, correlation coefficient (r), and R-squared (R²) values using your predictor and response data.
To use the calculator, provide a list of values for the predictor and the response, ensuring they are the same length, and then click the “Do Linear Regression” button.
Regression Line: y = + x
Goodness of fit:
Correlation r =
R-Squared =
Interpretation:
The y-intercept represents the mean of the response variable when the predictor variable is zero.
For each single unit increase in the predictor variable, there is an associated average change of in the response variable.
The R-squared value tells us that % of the variation in the response variable is predictable from the predictor variable.
Linear Regression Explanation
Linear Regression is a statistical method used to model the relationship between a dependent variable (response) and one or more independent variables (predictors). It is one of the simplest forms of regression analysis, commonly used to find the linear relationship between two variables.
Key Concepts
- Predictor Variable (X): The independent variable used to predict the outcome.
- Response Variable (Y): The dependent variable being predicted or explained.
- Regression Line: The straight line that best fits the data points on the plot.
- Slope (m): Indicates how much the response variable changes for a one-unit change in the predictor variable.
- Y-Intercept (b): The value of Y when the predictor variable (X) is zero.
Linear Regression Formula
The relationship between the predictor and response variables in linear regression is expressed using the equation of a line:
where \( Y \) is the predicted value of the response, \( X \) is the predictor, \( m \) is the slope of the line, and \( b \) is the y-intercept.
Steps to Perform Linear Regression
- Collect data for the predictor and response variables.
- Use the least squares method to estimate the slope and intercept of the regression line.
- Calculate the predicted values of the response variable for each value of the predictor.
- Evaluate the model's goodness of fit using metrics like the correlation coefficient (r) and R-squared (R²).
Goodness of Fit
Linear regression models are evaluated based on how well they fit the data:
- Correlation Coefficient (r): Measures the strength of the relationship between the predictor and response variables. A value close to 1 or -1 indicates a strong linear relationship.
- R-Squared (R²): Represents the proportion of the variance in the response variable that can be explained by the predictor variable(s). Higher values indicate a better fit.