H2 Maths correlation and regression formula sheet: product-moment correlation coefficient r, least-squares regression lines (y-on-x and x-on-y), interpolation vs extrapolation,...
Q: What does H2 Maths Notes (JC 1-2): 6.6) Correlation and Linear Regression cover? A: Product-moment correlation, least-squares lines, residual interpretation, and prediction limits for H2 Maths.
Before you revise Revisit scatter diagram basics and variance formulas so the transition to algebraic PMCC and regression is smooth. Keep a graphing calculator (GC) or spreadsheet handy to compute r and regression coefficients quickly.
The core idea is simple: Correlation measures association, not cause.
Use it as a working check: Regression predicts one variable from another, so decide which variable is the input before calculating.
Then go one layer deeper: Reliable prediction stays near the observed data range. Example: use Physics score to predict Maths score only if the Physics score is inside the original data range.
Concrete example: A strong link between Physics and Maths scores can help predict one from the other, but it does not prove that one subject caused the other score.
Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 6.6 is assessed in Paper 2 Section B (Probability and Statistics, 60 marks) and excludes hypothesis tests.
Formulas at a glance
Every result the 9758 syllabus expects you to recall, on one screen. The GC computes r and the regression coefficients directly - focus your memorisation on what each result means and when to use the correct regression direction. Worked examples for each appear in the sections below.
Correlation coefficient
Quantity
Formula
Product-moment correlation coefficient
r=(n−1)SxSy∑(xi−xˉ)(yi−yˉ)
Range
−1≤r≤1
Coefficient of determination
r2: proportion of variance in y explained by x
Regression lines
Line
Equation
Minimises
y on x
y−yˉ=b(x−xˉ), where b=rSxSy
Sum of squared vertical residuals ∑(yi−yi^)2
x on y
x−xˉ=b′(y−yˉ)
Fixed point
Both lines pass through (xˉ,yˉ)
-
Residuals and interpretation
Quantity
Formula / rule
Residual
ei=yi−yi^
Interpolation
Predicting within the observed x-range: reliable if the linear model is appropriate
Extrapolation
Predicting outside the observed x-range: unreliable
Core Definitions
Product-moment correlation coefficient r measures linear association between x and y; −1≤r≤1.
Least-squares regression line of y on x minimises ∑(yi−yi^)2
Sx and Sy are sample standard deviations of x and y
A scatter diagram plots bivariate data; a rising cloud indicates positive linear correlation. The least-squares regression line ŷ = a + bx minimises the sum of squared vertical residuals, and the product-moment correlation coefficient r (−1 ≤ r ≤ 1) measures how tightly the points cluster about it.
PMCC interpretation checkpoint
When a question asks you to comment on r, build the sentence in four parts instead of quoting the number alone.
Part to decide
What to look at
Sentence fragment
Common trap
Direction
Sign of r
positive or negative association
Saying "high" without saying the direction.
Strength
Distance of r from zero
weak, moderate, or strong linear association
Treating r=−0.92 as weak because it is negative.
Shape
Scatter or residual pattern
linear association if the pattern is roughly straight
Using r to describe a curved relationship.
Causation
Context of the variables
association only, not proof of cause
Writing that one variable causes the other.
Worked check: if r=−0.86 for revision hours missed and test score, write "There is a strong negative linear association between revision hours missed and test score." Do not write that missing revision caused the lower score unless the question gives causal evidence.
Misconception check: the sign tells you direction, while the distance from zero tells you strength. A negative value can still describe a strong association.
Computing r
For paired data (xi,yi), r=(n−1)SxSy∑(xi−xˉ)(yi−yˉ).
Example -- Physics vs Maths scores
Data (Physics,Maths) for 6 students: (68,72),(74,75),(65,69),(80,83),(70,73),(78,81).
Enter into GC lists and run LinReg(ax+b).
Output (rounded): r=0.984, xˉ=72.5, yˉ=75.5.
Strong positive linear association.
Regression Line and Prediction
Using the same data, the regression line of y on x (Maths on Physics) is
y=0.913x+9.34 (3 s.f.).
To predict Maths score when Physics = 76: substitute x=76, obtaining y≈78.7 (nearest whole number 79).
Only predict within the range of observed x (interpolation). Extrapolation is unreliable.
Prediction reliability checkpoint
For prediction questions, the arithmetic is only half the answer. Check whether the input value belongs to the data range before deciding how confidently to use the regression line.
define x and y
-> check observed x-range
-> substitute into the correct line
-> qualify the prediction
Prediction situation
What to check first
How to word the answer
Input value lies inside the observed data range
This is interpolation.
The prediction is reasonable if the linear model is appropriate.
Input value lies just outside the range
This is mild extrapolation.
The prediction is less reliable because it extends beyond the data.
Input value is far outside the range
This is unsafe extrapolation.
The regression line should not be trusted for this prediction.
Residual plot shows a curve or fan shape
The linear model may be unsuitable.
Even an in-range prediction should be treated with caution.
Worked check: if the Physics scores used to fit the model range from 65 to 80, predicting Maths score for Physics = 76 is interpolation. Predicting Maths score for Physics = 95 is extrapolation, so the numerical answer should be accompanied by a warning about reliability.
Misconception check: a high r value does not make every prediction safe. Regression reliability depends on the data range and whether a straight-line model is suitable.
Decision map - choose the regression direction first
Before pressing LinReg, translate the question sentence into input and output variables. The output is the variable you want to predict.
Question wording
Input variable
Output variable
Regression line to use
Predict Maths score from Physics score.
Physics score
Maths score
y on x, where x=Physics and y=Maths
Predict Physics score from Maths score.
Maths score
Physics score
y on x, but redefine x=Maths and y=Physics
Explain whether the two scores are associated.
neither variable is the input
neither variable is predicted
use r, not a prediction line
Misconception check: swapping the variables is not the same as rearranging the first regression equation. The least-squares line for predicting Maths from Physics minimises vertical errors in Maths. The line for predicting Physics from Maths is fitted again with the roles reversed.
Residuals and Coefficient of Determination
Residual: ei=yi−yi^; plot residuals to check linear model adequacy.
Coefficient of determination r2 gives the proportion of variance in y explained by x.
Example -- Interpretation
If r=0.82, then r2=0.6724. State: “About 67% of the variation in Maths marks is explained by Physics marks via the fitted linear model.”
r2 interpretation checkpoint
When interpreting r2, name the response variable and keep the statement tied to the fitted linear model.
Step
What to write
Why it matters
Common trap
Square the correlation
Convert r to r2, then to a percentage.
r2 is non-negative even when r is negative.
Saying a negative correlation gives a negative percentage explained.
Name the response variable
Say "variation in y" or name the actual predicted variable.
The interpretation follows the variable on the left of the regression line.
Saying "variation in both variables" without specifying the response.
Mention the model
Add "by the fitted linear model" or "by the linear relationship with x".
r2 summarises a linear fit, not every possible pattern.
Treating r2 as proof that the context has a causal explanation.
Leave the rest unexplained
State that the remaining percentage is not explained by the model.
It avoids overclaiming from a single statistic.
Calling the unexplained part "error" without checking residuals or context.
Worked check: if r=−0.70 for hours of sleep and reaction time, then r2=0.49. A careful sentence is: "About 49% of the variation in reaction time is explained by the fitted linear relationship with hours of sleep." The negative sign belongs in the direction of association, not in the percentage explained.
Misconception check: r2 does not say that 49% of reaction time is caused by sleep. It describes how much variation in the response variable is accounted for by the fitted straight-line model.
Residual plot checkpoint
After fitting a regression line, use the residual plot to check whether the linear model is still sensible.
Residual plot feature
What it suggests
What to write
Common trap
Points scattered randomly around 0
Linear model is reasonable.
There is no obvious pattern in the residuals, so a linear model is adequate.
Saying the original scatter plot has no pattern.
Curved pattern
Relationship may be non-linear.
A linear model may be unsuitable because residuals show systematic curvature.
Quoting a high r value and ignoring the curve.
Fan shape, with spread increasing
Variability changes with x.
Predictions become less consistent as x increases.
Treating all predictions as equally reliable.
One large isolated residual
Possible outlier.
Investigate that data point before relying on the fitted line.
Deleting the point without justification.
Worked check: if residuals are mostly positive for small and large x, but negative near the middle, the fitted straight line is missing a curved trend. The problem is not arithmetic; the model shape is wrong for the data.
Calculator Workflows
TI: LinReg(ax+b) returns a (gradient), b (intercept), r, and r2 when diagnostics are on.
Casio: REG mode →LR→AX+B provides coefficients and r.
Use residual lists to draw scatter plots of x vs residuals; randomness indicates a good linear fit.
Always state variables: “Let x=Physics score,y=Maths score.”
Exam Watch Points
Label axes and highlight whether you are predicting y from x or vice versa. The regression line of x on y is different.
Interpret r in words (“strong/weak, positive/negative”) and link back to context.
Check units: regression line must retain units of y on the left-hand side.
Do not claim causation; correlation only addresses association.
Mention interpolation vs extrapolation explicitly when commenting on prediction reliability.
Practice Quiz
Test your correlation interpretation and regression modelling in one sitting.
Quick Revision Checklist
Compute r and regression coefficients quickly with calculator support.
Write regression equations in the form y=ax+b and perform predictions.
Interpret r2 and residual plots qualitatively.
Explain interpolation vs extrapolation and avoid causal claims.
Want weekly guided practice on Correlation and Regression? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.
Common exam mistakes
Confusing the two regression lines: The line of y on x minimises vertical residuals; the line of x on y minimises horizontal residuals. Using the wrong line to make a prediction loses accuracy marks.
Claiming causation from correlation: Stating that a high r value means one variable causes the other is incorrect. Always describe the relationship as an association.
Extrapolating outside the data range: Predictions are reliable only for x-values within the observed range. Using the regression equation far beyond the data is flagged as unreliable extrapolation.
Forgetting to define variables: Marks are awarded for stating "let x = ... and y = ..." before writing the regression equation. Skipping this loses the context mark.
Interpreting r2 as a percentage without linking it to the variable: Write "r2=0.67 means about 67 percent of the variation in y is explained by the linear relationship with x
Frequently asked questions
Is there a formula sheet for H2 Maths correlation and regression? Yes - the "Formulas at a glance" section near the top of this page collects every result you need: the PMCC formula, both regression line equations and the (xˉ,yˉ) fixed point, and the interpolation vs extrapolation rule. Note that the GC computes r and the regression coefficients (a and b in y=ax+b) directly, so you do not need to evaluate the PMCC formula by hand. What you must supply yourself: choosing the correct regression direction (y on x vs x on y), interpreting r in context (direction, strength, no causation), interpreting r2 as a proportion of variance explained, and judging whether a prediction is interpolation or extrapolation.
Is correlation and regression in Paper 1 or Paper 2? Topic 6.6 is assessed in Paper 2 Section B (Probability & Statistics, 60 marks). Paper 1 is Pure Mathematics only.
Can I use the GC to find r and the regression equation? Yes - GC is expected. Use LinReg (TI) or REG → LR → AX+B (Casio) to obtain a, b, and r directly. You must still define variables, state the regression equation clearly, and interpret results in context.
Are correlation hypothesis tests included in the 2026 syllabus? No. The 2026 H2 Maths (9758) syllabus explicitly excludes hypothesis tests on the correlation coefficient. Focus only on computing, interpreting, and applying PMCC and the least-squares regression line.
Other H2 Maths formula sheets
Revising more than one topic? Grab the matching one-page formula sheet:
Next steps: Keep the H2 Maths notes hub on standby for mixed-paper practice that blends this regression module with sampling (6.4) and hypothesis testing (6.5).
