Correlation
Introduced| English |
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| correlation/ˌkɒrɪˈleɪʃn/ |
One number for a linear trend
- Correlation 相关性 $r$ is a single number for the strength and direction of a linear relationship.
- Its sign matches the direction: $r>0$ positive, $r<0$ negative.
- Its size measures tightness: near $\pm 1$ = tight line, near $0$ = no linear trend.
- $r$ turns "looks fairly strong" into a precise, comparable value.
Always between −1 and 1
- Correlation is bounded: $-1 \le r \le 1$, always.
- $r=+1$: a perfect increasing line; $r=-1$: a perfect decreasing line.
- $r=0$: no linear association (the points may still have a non-linear pattern).
- $r$ has no units and doesn't change if you switch which variable is $x$.
When r misleads
- $r$ measures linear strength only — a strong curve can give a middling $r$.
- A single outlier can pull $r$ up or down dramatically.
- So never report $r$ without first looking at the scatterplot.
- A high $r$ on a curved or outlier-driven plot is a false summary.
Correlation is not causation
- A large $r$ shows the variables move together, not that one causes the other.
- Ice-cream sales and drownings correlate — both rise in summer (heat is a lurking variable).
- Establishing cause needs an experiment, not just a high correlation.
- "Correlation does not imply causation" is the mantra of the whole unit.
$r$ is a linear measure and is not resistant. A curved relationship can have a small $r$ even though the variables are tightly related, and one outlier can inflate or deflate $r$ a lot. Always look at the scatterplot before trusting $r$ — and remember a high $r$ never proves causation.
Two datasets both have $r \approx 0.2$.
- Dataset A: a genuine weak, scattered linear trend — $r=0.2$ is honest.
- Dataset B: a strong U-shaped curve — the linear $r$ is near $0$, hiding a real relationship.
- Same $r$, totally different stories — which is why you must see the plot.
Correlation $r$ measures the strength and direction of a linear relationship, with $-1 \le r \le 1$ (sign = direction, magnitude = tightness, no units). It is misleading for curved data or with outliers, so always check the scatterplot — and correlation never implies causation.
Correlation and the point cloud
Raise the correlation and the points hug the line more tightly.
What is the largest possible value of a correlation coefficient r?
r ranges from −1 to 1, so the maximum is 1.
A correlation of r = 0 means there is definitely no relationship of any kind between the variables.
r = 0 means no LINEAR relationship; a strong curve can still exist.
Ice-cream sales and drownings both rise in summer, giving a high correlation. This is best explained by...
Summer heat drives both — correlation without causation.
Which can make r a misleading summary?
r is unitless; the real traps are curves, outliers, and skipping the plot.
Correlation r measures the strength and direction of a ___ relationship (one word).
r captures linear strength only.