Pearson correlation
Measure strength and direction of a linear sample relationship.
Pearson correlation is one of 2 association formulas in the statistics section of this library, and it is used at ap · university level.
Why pearson correlation works
The numerator is the covariance and the denominator rescales it by the spread of each variable, which cancels both sets of units. What is left is the average product of standardised deviations, and that quantity is mathematically trapped between -1 and 1, with the ends reached only by a perfect straight line.
What each symbol means
$r$ is unitless and lies between $-1$ and $1$.
Pearson correlation: when it holds
Both variables must vary; correlation does not establish causation and describes linear association.
When it stops applying
It measures straight-line association only, so a high value does not confirm a line is the right model. For x values 1 through 5 paired with their squares, the correlation is about 0.981 even though the true relationship is visibly curved. Always look at the scatterplot before trusting the number.
Pearson correlation: a worked example
$r$ near $1$ indicates a strong positive linear association.
The mistake to avoid
What people do: Treating a strong correlation as evidence that one variable causes the other.
Why it goes wrong: Two variables can move together because a third one drives both, or by pure coincidence in a small sample. Correlation orders the association; it says nothing about the direction of any influence.
Do this instead: Ask what else could explain the pattern, and remember that only a controlled experiment with randomised assignment supports a causal claim.
Pearson correlation: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $r$ is unitless and lies between $-1$ and $1$.
- Check the conditions before substituting. Both variables must vary; correlation does not establish causation and describes linear association.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most statistics slips.
Where this formula fits
- Subject
- Statistics formulas — 21 entries in this library
- Topic
- Association
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where pearson correlation comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Linear Regression — the lesson behind this formula: fit and interpret a linear relationship honestly.
- Statistics Calculator — check your substitution and the value it produces.
- Study statistics — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 21 statistics formulas — the full grouped reference, or the complete formula library.
Questions about pearson correlation
What does r squared add that r does not?
It converts the association into a share of variation accounted for. A correlation of 0.8 sounds like most of the story but corresponds to 0.64, meaning 64 percent of the variation in y is explained.
Does swapping which variable is x change the value?
No, the formula is symmetric in the two variables. That is a clue that correlation cannot indicate direction of cause, since it cannot even tell the two roles apart.
How strong does r have to be to matter?
It depends entirely on the field. In physics 0.9 might signal a problem with the equipment, while in social research 0.3 can be a genuinely important finding worth reporting.
Can r be exactly 1 for data that is not a straight line?
No. The value 1 is achieved only when every point sits exactly on a rising line, which is why an exact 1 in real data usually means one variable was computed from the other.