Read correlation with the scatter

Choose the response variable and appropriate regression direction. Check a scatterplot for curvature and outliers before trusting a correlation coefficient. Explain whether a prediction stays inside the observed range.

Calculate Pearson’s linear correlation coefficient

Look at the scatter before reading the coefficient. Describe the direction and strength of a linear pattern, then check for curvature and unusual observations. A coefficient near zero rules out neither a curved relationship nor a useful model of another form.

Pearson’s product moment correlation coefficient $r$ describes the strength and direction of linear association between paired numerical variables. Calculate it alongside the scatter, because a curved pattern is not summarised adequately by linear correlation.

$$r=\frac{\sum(x-\bar x)(y-\bar y)}{\sqrt{\sum(x-\bar x)^2\sum(y-\bar y)^2}}=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}},$$

where

$$S_{xy}=\sum xy-\frac{(\sum x)(\sum y)}n,\quad S_{xx}=\sum x^2-\frac{(\sum x)^2}n,\quad S_{yy}=\sum y^2-\frac{(\sum y)^2}n.$$

These are the centred and summary-data forms printed in MF27. Use the calculator directly when the paired observations are supplied; use the summary formula when only totals are available. The denominator requires both variables to vary, so $S_{xx}>0$ and $S_{yy}>0$. Otherwise $r$ is undefined, rather than zero.

Read the coefficient cases

For defined $r$, $-1\le r\le1$:

$r=-1$

All points lie on a line with negative gradient.

$-1<r<0$

Negative linear association: $y$ tends to decrease as $x$ increases.

$r=0$

Zero linear correlation; a strong curved relationship may remain.

$0<r<1$

Positive linear association: $y$ tends to increase as $x$ increases.

$r=1$

All points lie on a line with positive gradient.

The source’s $r<0$ and $r>0$ rows include the perfect endpoints; the separate rows above make the cases disjoint. “Tends to” describes the overall association, not every consecutive observation. The five source illustration roles are retained in Five coefficient cases.

Interpret the coefficient with its scatter

A strong correlation does not establish that one variable directly causes the other. The source compares a person’s right-foot size (horizontal) with left-foot size (vertical): its five-point rising schematic can reflect a shared factor such as genes. Its sentence “there is no causal relationship” is too general: correlation alone establishes neither the presence nor the absence of causation.

The three source calculator plots all report $r=0.86$, yet depict a roughly linear scatter, a curve, and a close trend with an unusual observation. Causation and equal-coefficient patterns preserves those distinct roles. The source supplies no raw coordinates for these three plots; the reconstructed illustrations must not be treated as datasets whose exact coefficient is 0.86.

Explain the effect of an unusual observation

An outlier can increase or decrease the correlation, or change its sign. In the source’s rising-scatter illustration, the point $(2,7)$ sits above the main trend; removing it gives a higher positive $r$. Outlier in a rising scatter preserves this example. Investigate unusual observations rather than deleting points just to increase $|r|$. The radiation example applies the formula and explains the effect of removing an observation without recalculating.

Construct a relationship that linear correlation misses

Use the paired data $x=-2,-1,0,1,2$ and $y=4,1,0,1,4$. Sketch the scatter and calculate r. Fit the regression of y on x and inspect its residuals. Explain why a zero coefficient does not settle whether there is a relationship.

After your attempt: compare the reasoning

Here $y=x^2$ exactly. Symmetry makes the centred cross-product sum zero while both variables vary, so $r=0$. The y-on-x fit is the horizontal line $y=2$. Its residuals are $2,-1,-2,-1,2$, showing curvature rather than an unstructured scatter about zero. The variables have a perfect quadratic relationship even though their linear correlation is zero. Inspect the scatter and model form alongside the coefficient.