Two variables form bivariate data when each individual provides a pair of values, written as $(x_1,y_1),(x_2,y_2),\dots,(x_n,y_n)$.
A scatter diagram (scatter plot) is usually the first representation because it lets you see whether an increase in $x$ tends to be associated with an increase or decrease in $y$.
Definition
Correlation
A measure of the association between two variables, describing whether they tend to increase together, decrease together, or show no consistent pattern.
Note
Correlation is useful because it supports generalization: if a relationship is strong and stable, you can often make cautious predictions using a line of best fit.
From the previous article you should know that covariance which measures whether $x$ and $y$ tend to deviate from their means in the same direction, and its common "population-style" formula is $$s_{xy}=\frac{\sum (x-\bar{x})(y-\bar{y})}{n}$$
To make the strength of the relationship meaningful, we divide by the spread (standard deviation) of each variable.
A standardized measure of linear correlation given by $$r=\frac{\frac{1}{n}\sum xy-\bar{x}\bar{y}}{\sigma_x\sigma_y},$$ where $\sigma_x$ and $\sigma_y$ are the standard deviations of $x$ and $y$.
Because we divide by $\sigma_x\sigma_y$, the coefficient $r$ has two key properties:
It is unit-free.
It always lies in the interval $-1\le r\le 1$.
Hint
Interpretation (linear patterns only):
$r=1$: points lie exactly on an upward-sloping straight line (perfect positive linear correlation).
$r=-1$: points lie exactly on a downward-sloping straight line (perfect negative linear correlation).
$r=0$: no linear relationship (there could still be a curved relationship).
Note
A value like $r\approx 0.8$ suggests a strong positive linear trend, but it does not mean the points lie on a line, and it does not prove that $x$ causes $y$.
How To Compute r From a Table of Values
For a set of $n$ pairs $(x,y)$, a practical workflow is:
Compute $\bar{x}$ and $\bar{y}$.
Compute $\sigma_x$ and $\sigma_y$ using $$\sigma_x=\sqrt{\frac{\sum (x-\bar{x})^2}{n}},\qquad \sigma_y=\sqrt{\frac{\sum (y-\bar{y})^2}{n}}$$
Compute $\frac{1}{n}\sum xy$.
Substitute into $$r=\frac{\frac{1}{n}\sum xy-\bar{x}\bar{{y}}}{\sigma_x\sigma_y}$$
On most calculators, $r$ is provided in the linear regression statistics menu.
Exam technique
Even when you use technology, always write an interpretation in context: direction (positive/negative) and strength (weak/moderate/strong).
Assessment often rewards communication, not just calculating $r$.
Example
Calculating $r$ and Linking It to a Scatter Plot
Consider the bivariate data:
$x$
12
14
15
17
19
$y$
19
20
22
23
25
First calculate key statistics (rounded where needed):
Standard deviations: $$\sigma_x\approx 2.417,\qquad \sigma_y\approx 2.136$$
So the correlation coefficient is $$r=\frac{6.08}{(2.417)(2.136)}\approx 0.965$$
The scatter plot shows points clustered close to an upward-sloping line, consistent with a strong positive linear correlation.
Tip
If your computed $r$ does not match the visual pattern, check for:
copying errors in the table
mixing up $n$ and $n-1$ (different courses/software may use different conventions)
or forgetting to divide $\sum xy$ by $n$.
Correlation Helps Describe Trends But Does Not Prove Causes
A strong correlation can justify drawing a line of best fit (by eye or using technology) and using it for interpolation, meaning prediction within the range of your data.
But there are important limitations.
Most importantly, correlation is not causation.
Two variables can be correlated because:
one causes the other,
they both depend on a third variable,
or the apparent relationship is coincidence in a small sample.
Also be careful with extrapolation (predicting outside the observed $x$ values), because the relationship may change beyond the data range.
Case study
In health data, a measure such as a "blood marker level" might correlate with a "taste score" for a group of patients.
Even if $r$ is fairly high, it would be unsafe to claim the blood marker causes taste changes without controlled experiments.
Correlation can suggest hypotheses, not confirm mechanisms.
Correlation In Human Contexts: Attitudes and Scores
Correlation is not only for physical measurements (mass, length, temperature).
It is also used with scores and survey data.
For example, students may have a social media news satisfaction score and a television news satisfaction score.
A scatter plot of the paired scores can show whether students who rate one medium highly also rate the other highly.
If the computed $r$ is close to 0 (or negative), you might conclude there is little (or an inverse) linear relationship between the two attitudes, which can lead to discussion about preferences and identity.
Active recall
Explain in words what $r=-0.7$ would look like on a scatter plot.
Why is $r$ unit-free while covariance is not?
Give one reason why $r\approx 0$ does not guarantee "no relationship".
In your own words, explain why extrapolation from a correlated data set can be risky.