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SECTION 06 / 8

Correlation: direction and strength without units

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Correlation standardizes covariance so its magnitude can be interpreted without measurement units. Pearson’s r summarizes a straight-line relationship.

Divide sample covariance by the two sample standard deviations, or use the equivalent sums formula.
Divide sample covariance by the two sample standard deviations, or use the equivalent sums formula.
r=sxysxsy=∑i(xi−xˉ)(yi−yˉ)[∑i(xi−xˉ)2][∑i(yi−yˉ)2]r=\frac{s_{xy}}{s_xs_y}=\frac{\sum_i(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\left[\sum_i(x_i-\bar{x})^2\right]\left[\sum_i(y_i-\bar{y})^2\right]}}

The sample n−1 factors cancel in the sums formula. If using covariance divided by standard deviations, use consistent sample definitions throughout, or consistent population definitions throughout. If either variable is constant, its SD is zero and correlation is undefined.

Read the sign and the absolute size

rInterpretation
+1Perfect positive straight-line relationship.
Near +1Strong positive linear association.
Near 0Little linear association; a curve may still exist.
Near −1Strong negative linear association.
−1Perfect negative straight-line relationship.

There is no universal cutoff that makes every correlation “strong” in every subject. Interpret the magnitude alongside the scatterplot, sample size and context. Correlation has no units and lies between −1 and +1.

Worked example: age and height

For the five pairs, the sum of squared age deviations is 10, the sum of squared height deviations is 677.2, and the sum of paired products is 82.

r=8210×677.2≈0.99645r=\frac{82}{\sqrt{10\times677.2}}\approx0.99645

This is a very strong positive linear relationship in these five observations. Keep full precision in intermediate values: using sample covariance 20.5, age SD √2.5 and height SD √169.3 gives the same result.

Worked example: car age and price

The car data have squared-deviation sums 10 and 10, and product sum −10. Thus r = −10/√(10×10) = −1. Every pair lies on Price = 7 − Age. This perfect example is a small teaching dataset, not a rule for all cars.

What changes r, and what does not?

Changing centimeters to meters, or adding the same constant to all measurements of one variable, leaves r unchanged. Multiplying just one variable by a negative number reverses the sign. Swapping x and y leaves r unchanged, although it changes a regression prediction problem. Outliers can strongly change r.

Zero correlation does not rule out a curve

For x = −2,−1,0,1,2 and y = 4,1,0,1,4, y equals x² exactly. The pattern is a perfect U-curve, but r = 0 because the negative and positive linear contributions cancel.

Spreadsheet and correlation matrix

For age in A2:A6 and height in B2:B6, =CORREL(A2:A6,B2:B6) calculates Pearson r. Keep rows paired and use the same complete cases. With several numerical variables, a correlation matrix lists r for every pair. It is symmetric, with 1 on the diagonal for nonconstant variables; it does not make a causal claim.

Try it yourself

Which is stronger as a linear association: r = −0.9 or r = +0.4?

Show explained answer

−0.9 has the stronger linear association because its absolute value is larger. The minus sign gives direction, not weakness.

Try it yourself

If r is 0.8 using kilograms, what is it after converting the same weights to grams?

Show explained answer

Still 0.8. Multiplying by a positive unit-conversion constant does not change correlation.

Try it yourself

Does r = 0.95 prove that x causes y?

Show explained answer

No. It describes strong linear association. Confounding, selection and other explanations remain possible.

Source: Lecture 4.7. Extra examples and diagrams are teaching material prepared for these notes.

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