Trend analysis
What does linear regression calculate?
Linear regression fits a straight line to paired numerical data. The model estimates how the dependent variable changes as the independent variable changes.
Model
Linear regression equation
Best-fit liney = mx + b
- m is the slope.
- b is the y-intercept.
- x is the predictor variable.
- y is the predicted response.
Worked example
Fit a line to 1,2; 2,4; and 3,6
- Mean of x = 2.
- Mean of y = 4.
- Calculated slope = 2.
- Calculated intercept = 0.
- Regression equation = y = 2x + 0.
Correlation
Interpret the correlation coefficient
The correlation coefficient r ranges from −1 to +1. Its sign indicates direction, and its magnitude indicates the strength of the linear association.
Positive r
Upward relationship
Larger x values tend to occur with larger y values.
Negative r
Downward relationship
Larger x values tend to occur with smaller y values.
Model fit
Interpret R squared
R squared describes the proportion of variation in y explained by the fitted linear relationship. A value closer to one indicates that the points lie more closely around the line.
Limitations
Avoid overinterpreting regression
- Correlation does not prove causation.
- A straight-line model may not fit curved relationships.
- Extreme points can strongly affect the slope and correlation.
- Predictions outside the measured x range may be unreliable.
- Units and experimental context still matter.
Graphing
Plot the data before interpreting the model
A scatter graph helps reveal curvature, clusters, anomalies, or other patterns that summary statistics alone may hide.
Review the Tables and Graphs Guide and the Graphing Scientific Data Worksheet.
Experimental interpretation
Connect the trend to the research question
Use the regression equation, direction, strength, anomalies, and experimental limitations when describing the evidence.
Follow the Experimental Results Analysis Guide for a complete interpretation workflow.
Questions and answers
Linear regression FAQ
What does the slope represent?
The slope represents the predicted change in y for each one-unit increase in x.
What does the intercept represent?
The intercept is the predicted y value when x equals zero, although it may not always have a meaningful experimental interpretation.
What does the correlation coefficient mean?
The correlation coefficient r describes the direction and strength of a linear relationship, with values closer to minus one or plus one indicating a stronger linear association.
What does R squared mean?
R squared is the proportion of variation in y explained by the fitted linear model.