# Correlation coefficient

The correlation r measures the strength and direction of a straight-line link between two numerical variables. It runs from -1 (perfect downhill line) to +1 (perfect uphill line). Zero means no straight-line link.

Statistics (university) · Part 08: Correlation and regression · Card 1 of 9 · Last updated 2026-10-19

## Analogy

Tall parents tend to have tall children, but not always. Correlation puts a number on how closely the two go together.

## Example

**Computing r**

x: 1,2,3,4,5  y: 2,4,5,4,5
Means: 3 and 4
Sum of dx x dy = 6
Sxx = 10, Syy = 6
r = 6 / square root of 60 = 0.77

## In practice

Used to check whether study time, price or temperature moves with another measure.

## Practice

### Flashcards

- **What range can the correlation r take?** From -1 to +1.

### Interview questions

**For the points (1,2), (2,4), (3,5), (4,4), (5,5), find the least squares line and R-squared.**

- Means 3 and 4
- Slope Sxy / Sxx = 6 / 10 = 0.6
- Intercept = 4 - 1.8 = 2.2
- R-squared = 0.60
- Residuals sum to zero

## Questions people ask

### What does a correlation of 0.77 mean?

It means a fairly strong uphill straight-line link: as x goes up, y tends to go up. It does not mean 77 percent of anything. Squaring r gives 0.6, the share of the variation in y explained by the line.

### Can the correlation be zero when two variables are strongly related?

Yes. Correlation only detects straight-line links. A perfect U-shaped curve, such as y equals x squared over a symmetric range, can have r equal to zero. Always plot the data first rather than trusting r alone, because a single number can hide the shape of the data.

## Related concepts

- [Least squares regression line](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-8/statistics-university-part-8-concept-2)
- [R-squared](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-8/statistics-university-part-8-concept-5)

Web page: https://studyvisually.com/learn/science/statistics-university/statistics-university-part-8/statistics-university-part-8-concept-1
Study this card in the app: https://studyvisually.com/app/learn?category=statistics-university&card=statistics-university-part-8-concept-1