MATH2209 Chap.10 Regression and Pearson Correlation
Regression and Pearson Correlation
Regression and correlation summarise linear relationships between two quantitative variables. A scatterplot comes first because curvature and influential observations can make one coefficient misleading. Correlation records unit-free direction and strength, while regression assigns explanatory and response roles to predict mean response.
This chapter interprets slope with units, calculates fitted values and residuals, and limits prediction to the observed domain without claiming causation from association.
What this chapter covers
- 01
Explanatory and response variables
- 02
Scatterplot interpretation
- 03
Least-squares fitted lines
- 04
Slope and intercept
- 05
Fitted values and residuals
- 06
Pearson correlation and linear association
Worked example · free
Predict mass from light exposure
- 1Fitted mass = 12 + 1.8 x 6 = 22.8 g.
- 1Residual = observed - fitted = 25.0 - 22.8 = 2.2 g.
- 1The positive residual places the observation above the fitted line.
- 1Each extra light hour is associated with 1.8 g higher predicted mean mass within the studied range.
Key terms
- Regression Slope
- The predicted change in mean response for a one-unit increase in the explanatory variable.
- Residual
- Observed response minus the response predicted by the fitted model.
- Pearson Correlation
- A unit-free measure of the direction and strength of linear association.
Regression and Pearson Correlation FAQ
Can correlation be near zero when variables are related?
Yes. Pearson correlation measures linear association. A strong curved relationship can have positive and negative local slopes that cancel to near zero. Inspect the scatterplot before using correlation as a summary.
Why does residual sign matter?
Residual equals observed response minus fitted response. Positive values place points above the line; negative values place them below. Reversing the subtraction changes every diagnostic sign and leads to incorrect interpretation.
When is a regression prediction unsafe?
Prediction is unsafe far outside the observed explanatory-variable range because biological relationships may plateau, reverse or become impossible. A straight equation continues mathematically, but evidence for it ends with the data domain used to fit the model.
What evidence should accompany a correlation?
Show the scatterplot and describe direction, form, strength and unusual observations before reporting Pearson r. A coefficient near zero can hide curvature, separate clusters or opposing local trends, while one influential point can dominate a linear summary. For regression, retain the observed x range, interpret slope with both variable units and calculate residuals as observed minus fitted.
Inspect residuals for curvature, changing spread and isolated gaps. Explain that correlation is unit-free and symmetric, whereas regression assigns explanatory and response roles. Neither establishes causation without design support. Any prediction should identify whether it is interpolation or extrapolation and should not extend biological meaning merely because the fitted equation returns a value.
Assessment move
For each equation, label x and y with units before substituting. Identify the explanatory variable, response variable, observed range and scientific question. Draw the scatterplot before calculating a coefficient. Describe direction, form, strength and unusual observations in that order, and decide whether one straight line is a reasonable summary.
Create contrasting sketches with a tight positive line, a weak negative cloud, a U shape, two clusters and one influential point. Predict how Pearson r behaves in each case and explain why a strong nonlinear relationship can still have r near zero. Practise coefficient language separately. Interpret the intercept only when x=0 is meaningful and within a defensible context.
Interpret the slope as the change in predicted mean response for a one-unit increase in x, retaining both units and using association language. Calculate fitted values by substitution, then residuals as observed minus fitted. Check the sign visually: a point above the line has a positive residual and one below has a negative residual. Build a small residual plot and look for curvature, changing spread and isolated gaps.
For model diagnostics, connect each pattern to a consequence. Curvature challenges the linear mean structure; a fan shape challenges constant variability; dependence can arise from repeated observations; and extrapolation extends beyond the evidence even when the fitted equation returns a number. Do not delete an influential point without investigating its origin and scientific relevance.
Pair correlation with regression questions using the same data. Explain that r is dimensionless and symmetric in x and y, while the regression slope has units and changes when predictor and response roles are exchanged. Neither coefficient proves causation. End every prediction with the x value and whether it lies inside the observed range.
Report uncertainty when supplied and distinguish a fitted mean response from an individual future outcome. Your final drill is to audit a published-looking line using the plot, residuals, units, range and design before accepting its p-value.
Working through Regression and Pearson Correlation in MATH2209? Sia is AskSia’s AI Statistics tutor — ask any MATH2209 Regression and Pearson Correlation question and get a clear, step-by-step explanation grounded in how MATH2209 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.