Exploring Two-Variable Data

Scatterplots & Correlation

Describing the form, direction, and strength of a relationship, and computing the correlation coefficient.

Intermediate20 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Describing a Relationship Before Measuring It

A car's age (in years) and its resale value (in thousands of dollars): (1,22), (2,19), (3,17), (4,14), (5,11), (6,9). Before reading on: just glancing at these pairs, does resale value seem to rise or fall as a car gets older, and does the pattern look close to a straight line or more scattered?

Definition — Describing a Scatterplot: Form, Direction, Strength

A scatterplot's form describes its overall pattern (linear, curved, or no clear pattern), direction describes whether y tends to increase or decrease as x increases (positive or negative), and strength describes how closely the points follow that pattern. The correlation coefficient r quantifies strength and direction numerically, ranging from −1 (perfect negative linear relationship) to +1 (perfect positive linear relationship), with 0 meaning no linear relationship.

r = \frac{\sum (x-\bar{x})(y-\bar{y})}{\sqrt{\sum (x-\bar{x})^2 \sum (y-\bar{y})^2}}

Worked Example — Computing the Correlation Coefficient

With x̄ = 3.5 and ȳ ≈ 15.33: Σ(x−x̄)(y−ȳ) = −46, Σ(x−x̄)² = 17.5, Σ(y−ȳ)² ≈ 121.33. r = −46/√(17.5 × 121.33) ≈ −46/46.08 ≈ −0.998. This confirms what the scatterplot suggests visually: the form is linear, the direction is negative (value drops as age increases), and the strength is very strong — r is extremely close to −1.

Graph Visualizer

Trace modePoint mode (click graph to plot)

Show x^2

Color

Expression

Domain & range

x min

x max

y min

y max

Precision (decimal places)2

Evaluate a point

Evaluate at x =

x^2 = 0

Tip

A correlation near −1 or +1 says the points hug a straight line closely — it says nothing about the line's steepness, which depends on the actual scale of the variables, not on r itself.

Common Mistakes

Interpreting r = −0.998 as 'a weak relationship because it's negative.'
The sign of r describes direction, not strength — a value close to −1 (like −0.998) is just as strong a relationship as a value close to +1; only how close r is to 0 indicates weakness.
Assuming a high |r| means age causes the drop in value.
Correlation measures the strength of a linear association only — it doesn't by itself establish that one variable causes changes in the other, even when the relationship is very strong.

Key Takeaways

A scatterplot's form, direction, and strength describe a relationship before any number is computed.
The correlation coefficient r ranges from −1 to +1, with its sign showing direction and its distance from 0 showing strength.
A strong correlation, however close to ±1, never by itself establishes causation.

Summary

Correlation measures how strong a linear relationship is. The next lesson fits an actual line to this same data and interprets what it means.

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