BSAN2201 Chap.5 Correlation and Linear Regression
Correlation and Linear Regression
Correlation summarises the strength and direction of a linear relationship, with a coefficient from minus one to plus one. Its sign gives direction and its absolute value describes strength, but neither establishes causation. Simple linear regression assigns a predictor X and target Y, then fits the equation Y = b0 + b1X.
The intercept is the predicted value of Y when X is zero, while the slope is the predicted change in Y for a one-unit increase in X. Interpretation requires units, a check against the plotted direction and a warning about extrapolation beyond the observed data.
The scatterplot remains part of the analysis because clusters, curves and influential observations can make one coefficient misleading.
The fitted line describes the expected target value, not every outcome, and its algebra should not be projected far beyond the sample without justification. A slope is an association unless the design supports causal interpretation.
The chapter therefore uses a fixed answer sequence: identify variables and units, inspect direction, write the equation, substitute the predictor, calculate the fitted target, interpret the slope and state the evidentiary limit.
What this chapter covers
- 01
Correlation direction and strength
- 02
Scatterplots and outliers
- 03
Predictor and target variables
- 04
Intercept and slope meaning
- 05
Substitution with units
- 06
Causation and extrapolation limits
Worked example · free
Predict revenue from training
- +1Substitute 8 into the equation: 42 + 3.5(8).
- +1Multiply 3.5 by 8 to obtain 28.
- +1Add 42 and 28 to predict 70 thousand dollars.
- +1Interpret 3.5 as an expected increase per additional thousand dollars of training, without claiming causation.
Key terms
- Correlation coefficient
- A measure from minus one to plus one of linear relationship direction and strength.
- Dependent variable
- The target variable predicted or explained in a regression model.
- Independent variable
- A predictor used to estimate the target variable.
- Regression intercept
- The predicted target value when the predictor equals zero.
- Regression slope
- The predicted target change for a one-unit predictor increase.
Correlation and Linear Regression FAQ
What does a negative correlation mean?
It means the variables tend to move in opposite directions: higher values of one accompany lower values of the other. The sign says nothing about causal direction, and the absolute value should be interpreted with the scatterplot because outliers or curves can mislead.
How should I interpret a regression slope?
State the predicted change in the target for a one-unit increase in the predictor, include both units, and keep the interpretation within the fitted range. Use language such as associated with unless the research design supports a causal claim.
Why can extrapolation be risky?
A linear equation continues indefinitely, but the business relationship may have capacity limits, thresholds or saturation outside the observed predictor range. A substitution can be arithmetically correct while the resulting prediction is poorly supported by evidence.
What should I check before using a fitted prediction?
Confirm that the predictor uses the expected units, lies within or near the observed range and comes from a population comparable with the fitting data. Inspect the scatterplot, unusual points and residual uncertainty. Then ask whether the predicted difference is large enough to change the business decision.
Assessment move
Practise the same four-line answer repeatedly: write the equation, substitute the predictor, calculate the target and interpret the slope in units. Sketch the expected line direction before calculating so a sign error is visible. For every correlation statement, add one sentence explaining why association alone does not establish an intervention effect.
Use invented datasets with different scales so units become automatic rather than an afterthought. Mark the observed predictor range on each scatterplot and label any requested value outside it as extrapolation. Remove and restore one verified extreme point to see how the fitted line changes, then report whether the business conclusion depends on it.
Finish with a directional audit: increasing X must move the prediction according to the sign of the slope. If it does not, recheck the equation, coding and selected columns. Create matched questions that use the same equation but change the units, sign and requested predictor. Estimate the direction before substitution, then compare the calculated change with the sketch.
Interpret the intercept only after checking whether zero is meaningful and represented. For each fitted value, add the model population, period and observed range. Next, write one plausible confounder and the evidence needed for a causal claim. This repeated structure separates arithmetic correctness, predictive support and intervention logic so none is smuggled into the others.
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