Residuals and the Correlation Coefficient

A residual is the model’s miss

A line of best fit predicts a value for every $x$. The actual data rarely lands exactly on it, and the gap has a name.

$$\text{residual} = \text{actual} - \text{predicted}$$

A line predicts test scores from hours studied: $\hat{y} = 7x + 32$. A student who studied $5$ hours scored $70$.

$$\text{predicted} = 7(5) + 32 = 67 \qquad \text{residual} = 70 - 67 = 3$$

Residual	Where the point sits	Reads as
positive	above the line	did better than predicted
zero	on the line	exactly as predicted
negative	below the line	did worse than predicted

The order matters. Actual minus predicted, always. Reversing it flips every sign and turns “above the line” into “below”.

The hat on $\hat{y}$ marks a predicted value, keeping it separate from the $y$ that was actually measured.

The residual plot

Calculate the residual for every point, then plot each one against its $x$. That is a residual plot, and it answers a question the scatter plot cannot answer by eye: was a straight line the right model at all?

Residual plot	Conclusion
random scatter around zero	a linear model fits
a curved pattern	the data is curved; a line is the wrong model
a fan that widens	the line’s errors grow with $x$

Random is good. If the line captured the trend, what is left over is noise, and noise has no shape. A pattern in the residuals is trend the line failed to capture.

That U shape means the data bends. A straight line cuts under it at both ends and over it in the middle, so the residuals come out positive, then negative, then positive again. The fix is a different model, often a quadratic or an exponential, not a better line.

Why a curve can hide in a scatter plot

A gently curved cloud of points can look close enough to straight on a scatter plot. The line runs through the middle and seems to fit.

The residual plot takes the trend away and magnifies what remains. Small, systematic misses that were invisible beside a large rising trend become the only thing on the graph.

That is why the residual plot is checked after fitting a line, even one that looked good. The scatter plot shows whether there is a trend; the residual plot shows whether you described it correctly.

The correlation coefficient

The correlation coefficient, $r$, summarizes how tightly data follows a straight line in a single number.

$$-1 \le r \le 1$$

$r$	Meaning
$1$	perfect positive line — every point on a rising line
close to $1$	strong positive linear relationship
close to $0$	weak or no linear relationship
close to $-1$	strong negative linear relationship
$-1$	perfect negative line

The sign is direction. The distance from zero is strength. An $r$ of $-0.9$ is a stronger relationship than $r = 0.6$, because $0.9$ is farther from zero. The minus sign only says the line runs downhill.

$r$ has no units and does not change if you switch units, say from inches to centimeters. It measures the shape of the cloud, not the size of the numbers.

Computing $r$ by hand is long, and in practice a graphing calculator or a spreadsheet’s CORREL function produces it. The skill that matters is reading it.

What r cannot tell you

Claim	Why $r$ does not support it
the relationship is linear	a curve can produce a large $r$ over a short range
there is no relationship	a perfect U shape can give $r$ close to $0$
one causes the other	$r$ says nothing about why the points line up

A large $r$ with a curved residual plot means the relationship is strong and not linear. Always pair $r$ with a look at the residuals.

And a strong correlation is still only a correlation. Correlation is not causation: a third, lurking variable can move both quantities together.

Worked examples

Common mistakes

Practice problems