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Statistics & Data Science · pattern F, Dataset

Pearson's r, with the sums it comes from.

Paste your x and y pairs, one per line, for Pearson's correlation coefficient, r², and the covariance sums a marker expects to see written out.

Inputs

Sample or population

Divide by n − 1 for a sample drawn from a wider population; by n only when the data is the entire population.

The formula used

r = Σ(x − x̄)(y − ȳ) ÷ √[Σ(x − x̄)² × Σ(y − ȳ)²]

One pair per line, separated by a comma, tab or space. Lines that do not contain two numbers are ignored.

Pearson's r

0.9949

n = 8 pairs · r = 0.9949 · r² = 0.9899 — strong linear association

Slope, b
1.55556
Intercept, a
23.00000
r
0.99494
r²
0.98990
Mean of x
22.500
Mean of y
58.000
Σ(x−x̄)(y−ȳ)
588.000
Σ(x−x̄)²
378.000

Your data — accent bars sit beyond one deviation

41.0073.00

Worked steps

  1. 1b = Σ(x−x̄)(y−ȳ) ÷ Σ(x−x̄)² = 588.000 ÷ 378.000 = 1.55556

    the slope is the covariance of x and y over the variance of x

  2. 2a = ȳ − b·x̄ = 58.000 − 1.55556·22.500 = 23.00000

    the line passes through the point of means

  3. 3r = 0.99494, r² = 0.98990

    r² is the share of variation in y explained by x

Correlation is a statement about a straight line.

Pearson's r measures how closely the points follow a straight line, so a perfect parabola can return an r near zero while being entirely determined. Always plot the data before quoting the coefficient. And r says nothing about which variable causes the other, or whether a third variable drives both — the standard caution exists because the mistake is genuinely easy to make in a discussion section.

Questions about correlation

What counts as a strong correlation?
It depends on the field. In physics 0.9 may be disappointing; in psychology 0.4 can be a substantial finding. Report r, r², n and a plot rather than a verbal label.
What does r² mean?
The proportion of variation in y that the linear relationship with x accounts for. An r of 0.7 gives r² of 0.49, so about half the variation is explained and half is not.
When should I use Spearman instead?
When the relationship is monotonic but not linear, or when the data is ordinal or has influential outliers. Spearman correlates the ranks rather than the values.

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