Correlation and Regression

Correlation and regression are the H2 Maths tools for describing the linear relationship between two variables. A scatter diagram gives an immediate visual impression of type (positive, negative, or none) and strength; the product-moment correlation coefficient $r$ quantifies this precisely, and the least-squares regression line lets you make estimates — provided you use it carefully.

What you need to know

• Scatter diagram: plot one variable against the other. The diagram reveals the type and approximate strength of any linear relationship and exposes outliers. Always label both axes.
• Product-moment correlation coefficient: $r$ is obtained from the GC and satisfies $-1 \le r \le 1$. Values close to $+1$ or $-1$ indicate strong positive or negative linear correlation respectively; values close to $0$ indicate little or no linear correlation (a clear non-linear pattern may still exist).
• Interpreting $r$: comment on both sign (direction) and magnitude (strength). A value such as $r = 0.95$ means strong positive linear correlation; $r = -0.3$ means weak negative linear correlation. Correlation does not imply causation.
• Least-squares regression line of $y$ on $x$: the line $\hat{y} = a + bx$ that minimises the sum of squared vertical residuals. Its coefficients are read from the GC. The line always passes through $(\bar{x}, \bar{y})$ and can be written

• Choosing the correct regression line: use $y$ on $x$ when estimating $y$ from a given $x$; use $x$ on $y$ when estimating $x$ from a given $y$. Do not rearrange the $y$-on-$x$ line to estimate $x$.
• Reliability of estimates: estimation within the data range (interpolation) is generally reliable when $|r|$ is close to 1. Estimation outside the data range (extrapolation) is unreliable because the linear model may not hold beyond the observed values.

Worked example 1 — Interpreting a given correlation coefficient

A study records the number of hours of revision $x$ and the score $y$ (out of 100) for 12 students. The GC gives $r = 0.88$.

Comment on the correlation between hours of revision and score, and state one limitation of this conclusion.

Solution.

Since $r = 0.88$ is positive and close to $1$, there is a strong positive linear correlation between hours of revision and score: students who revised for more hours tended to score higher.

However, correlation does not imply causation. Other factors (prior ability, quality of resources, sleep) may also influence scores. The conclusion applies only to the range of revision hours observed in the sample; extrapolating beyond that range may not be valid.

Worked example 2 — Using the regression line to estimate a value

The heights $x$ (in cm) and shoe sizes $y$ of 10 adults are recorded. Summary statistics give $\bar{x} = 170$, $\bar{y} = 8.2$, and the GC produces the regression line of $y$ on $x$:

The correlation coefficient is $r = 0.91$. The heights in the sample range from 155 cm to 185 cm.

(a) Estimate the shoe size of an adult of height 175 cm, and comment on the reliability of this estimate.

(b) A colleague suggests using this line to estimate the shoe size of a child of height 120 cm. Comment on this suggestion.

Solution.

(a) Substituting $x = 175$:

The estimated shoe size is 8.8. Since $r = 0.91$ indicates strong positive linear correlation and $x = 175$ cm lies within the observed data range (155 cm to 185 cm), this is an interpolation and the estimate is reliable.

(b) A height of 120 cm is well outside the observed range, so this would be an extrapolation. The linear model may not hold for children, and the estimate would be unreliable. The colleague’s suggestion is not appropriate.

Worked example 3 — Verifying the line passes through $(\bar{x}, \bar{y})$

Using the data from Worked example 2, verify that $(\bar{x}, \bar{y}) = (170, 8.2)$ lies on the regression line.

Solution.

Substitute $x = \bar{x} = 170$ into the regression equation:

This does not exactly equal $\bar{y} = 8.2$ due to rounding in the stated coefficients. In an exam, the GC’s unrounded coefficients will satisfy $\hat{y} = \bar{y}$ exactly when $x = \bar{x}$. This property is a useful check that you have used the correct regression line.

Common mistakes

• Rearranging the $y$-on-$x$ line to estimate $x$. The regression line of $y$ on $x$ minimises vertical residuals and is not the same as the line of $x$ on $y$. To estimate $x$ from a known $y$, obtain and use the separate $x$-on-$y$ regression line from the GC.
• Concluding causation from correlation. A high $|r|$ shows a strong linear association, not that one variable causes the other.
• Treating $r \approx 0$ as “no relationship”. It means little or no linear relationship; a strong non-linear pattern (e.g. quadratic) can still give $r$ close to $0$.
• Extrapolating without comment. Any estimate outside the observed data range must be accompanied by a statement that it is an extrapolation and therefore unreliable.