Scatter plot maker

The trend line

y = a + bx, b = Σ(x − x̄)(y − ȳ) ÷ Σ(x − x̄)², a = ȳ − b·x̄ r = Σ(x − x̄)(y − ȳ) ÷ √[Σ(x − x̄)² × Σ(y − ȳ)²], R² = r²

The least-squares line is the one that makes the sum of squared vertical distances from the points as small as possible. It always passes through the point of means (x̄, ȳ). A residual is a point's vertical distance from the line, y − ŷ, and the residual standard error sₑ = √(Σ(y − ŷ)² ÷ (n − 2)) is their typical size.

Worked example

The default data are hours studied (x) and exam score (y) for ten students: (1, 52), (2, 55), (2, 61), (3, 60), (4, 66), (5, 70), (5, 48), (6, 75), (7, 79) and (8, 84).

• Means: x̄ = 43 ÷ 10 = 4.3 hours and ȳ = 650 ÷ 10 = 65 marks.
• Sums of squares: Σ(x − x̄)² = 48.1, Σ(y − ȳ)² = 1,282 and Σ(x − x̄)(y − ȳ) = 198.
• Standard deviations: sₓ = √(48.1 ÷ 9) = 2.312 hours and s_y = √(1,282 ÷ 9) = 11.94 marks.
• Slope b = 198 ÷ 48.1 = 4.116 and intercept a = 65 − 4.116 × 4.3 = 47.30, so the line is y = 47.30 + 4.116x.
• r = 198 ÷ √(48.1 × 1,282) = 0.797, a strong positive correlation, and R² = 0.636.

Each extra hour of study goes with about 4.1 more marks on average. The residual standard error is 7.64, so 2sₑ = 15.28. The student at (5, 48) is predicted to score 47.30 + 4.116 × 5 = 67.88 and actually scored 48, a residual of −19.88, or −2.60 sₑ. That is the one flagged point. Take it out and r rises to 0.988 with R² = 0.976, which shows how much a single unusual case can drag on the correlation in a small sample.

How to read a scatter plot

Look for five things, roughly in this order.

• Direction. Points rising from left to right show a positive association; falling points show a negative one.
• Form. Do the points follow a straight band, a curve, or no pattern at all? r and the straight trend line only describe the straight-line part.
• Strength. The tighter the points hug a line, the stronger the relationship. As a guide, |r| above 0.7 is strong and below 0.3 is weak.
• Outliers. A point far from the overall pattern can be a data-entry error or a genuinely unusual case. Check which before deciding what to do with it.
• Clusters. Separate clumps often mean two groups are mixed together, such as two classes or two machines, and each group may have its own trend.

Correlation is not causation

A clear pattern shows that two variables move together, not that one drives the other. The link can run the other way, or a third variable can drive both: students who study longer may also sleep better or have more support at home. Ice-cream sales and drownings both rise in summer, but heat is the common cause. Only a controlled experiment, or careful reasoning about confounders, can support a causal claim.

When to add a trend line

Add the least-squares line when the points form a roughly straight band and you want to summarise the rate of change or make a rough prediction inside the range of your x values. Leave it off when the pattern is clearly curved, when the data fall into separate clusters, or when you only want to show the raw data. Do not extend the line far beyond the smallest and largest x: the trend may not hold there. With only two points the line fits them exactly and tells you nothing about the fit.

Common mistakes

• Swapping the axes. The regression of y on x is not the same line as x on y, although r is the same either way.
• Reading r = 0 as “no relationship”. A strong U-shaped pattern can have r close to zero.
• Deleting flagged points without a reason. Remove a point only if it is a proven error or clearly belongs to a different population, and say so.

For a significance test on r, use the correlation coefficient calculator; for standard errors and predictions, the linear regression calculator; to study every residual, the residual calculator; and for the unscaled measure of joint spread, the covariance calculator.