Correlation
Do taller people really weigh more? Does an extra hour of revision nudge up your marks? Whenever you
suspect two things rise or fall together, correlation puts a single number on exactly how tightly they
are linked — the kind of number that fills sports analytics, medical studies and market research.
A scatter plot
shows you a relationship; the correlation coefficient
r puts a single number on it. It squeezes the whole question "how tightly
do these two variables move together?" into one value between -1 and
+1 — where +1 is a perfect upward line,
-1 a perfect downward line, and 0 no linear
relationship at all.
That is enormously useful — one number you can compare, rank, and report. It is also, as you will
see, one number that can lie to you spectacularly if you never looked at the scatter first. Both
things are true, and that tension is the whole story of this page.
Sign and magnitude
r lives in
-1 \le r \le 1,
and measures the strength and direction of the
linear relationship between two variables.
- The sign is the direction: r > 0 for a rising cloud, r < 0 for a falling one.
- The magnitude is the tightness: |r| near 1 means the dots hug a line; near 0 means a loose, shapeless scatter.
So r = 1 and r = -1 are perfect straight
lines (up and down); r = 0 is no linear trend at all. Read the two halves
separately — sign tells you which way, magnitude tells you how tightly — and you
can decode any reported r.
Where the number comes from
Standardise each variable into z-scores — subtract the mean and divide by the
standard deviation — so both axes are measured in the
same unitless scale. Then r is simply the average product of
the paired z-scores:
r = \frac{1}{n}\sum_{i=1}^{n} z_{x_i}\, z_{y_i}, \qquad z_{x_i} = \frac{x_i - \bar x}{s_x},\quad z_{y_i} = \frac{y_i - \bar y}{s_y}.
Read the sign off the products: a point that is above average in both
x and y contributes
(+)(+) > 0; below average in both gives
(-)(-) > 0. Points that match this way push
r up; points that disagree (high
x, low y) pull it down. When agreements
and disagreements cancel, r \approx 0.
Loosen the cloud
Each dot starts on the perfect line y = x and is then nudged off it by
a fixed amount times the noise dial. With no noise the points are collinear and
r = 1; as you add noise the cloud fattens and r
slides toward 0. The live readout recomputes
r from the points on screen.
Three worked reads
You rarely compute r by hand — you interpret one someone hands you. Here
is how to talk about three of them.
1. Estimating from a picture. A scatter of study-hours versus exam-score rises
clearly to the right and the dots sit fairly close to a straight line, with a little scatter. That
is a strong positive linear cloud, so a sensible estimate is around
r \approx 0.8. Not 1 (the dots aren't
perfectly collinear), but comfortably strong and positive.
2. Reading a reported value. A study reports
r = 0.8: strong and positive — the two variables rise together
and the cloud is tight. Compare r = -0.3: the sign says the relationship
is negative (one falls as the other rises), but the small magnitude says it is weak
— a loose, barely-tilted blob. Sign and magnitude, read separately, every time.
3. When a perfect relationship gives r \approx 0. Take
y = x^2 with x symmetric around zero — a
perfect, fully predictable parabola. Compute r and you get roughly
0: the rising right half and falling left half contribute opposite-sign
products that cancel. The relationship is flawless, yet r is blind to it,
because r only measures linear association. So
r = 0 means "no line", never "no relationship".
This is the single most important warning in all of statistics: correlation is not
causation.
Ice-cream sales and drowning deaths are strongly correlated — both climb in summer — but neither
causes the other. A lurking variable, the hot weather, drives both. A high
|r| means only that two things move together; it says nothing
about why. Confusing the two underlies a staggering number of bogus health claims, policy
blunders, and "X causes Y" headlines. Before you believe any "X causes Y", ask: could a third
variable drive both? Could it be pure coincidence? Could the arrow point the other way? A tight
correlation is a question, not an answer.
A gleeful website called Spurious Correlations collects pairs of totally unrelated
things that happen to track each other almost perfectly. US per-capita cheese consumption
marches in near-lockstep with the number of people who died tangled in their bedsheets.
The number of Nicolas Cage films released each year rises and falls with the number of
swimming-pool drownings. These correlations are enormous — and utterly meaningless.
The lesson isn't magic, it's arithmetic: rummage through enough variables and some pairs will line
up sky-high purely by chance. It also highlights a real distinction. Correlation is
symmetric — it just says "these two move together", with no direction. When you instead
want an asymmetric "use x to predict
y", you need a
regression line,
which treats the two variables quite differently.
The one-line summary
- r \in [-1, 1] measures the strength + direction of a linear relationship.
- It is the average product of z-scores: r = \frac{1}{n}\sum z_{x_i} z_{y_i}.
- Sign = direction; |r| near 1 = tight, near 0 = scattered.
- r = 0 rules out a line, not a curve — and correlation is never proof of cause.
See it explained