Measuring the Lean
The correlation coefficient r condenses the direction and tightness of a linear relationship into a single number between minus one and one. · 13 min
Last folio you called a relationship strong or weak by eye. But different people, looking at the same cloud, will disagree about where 'strong' ends and 'moderate' begins. Statistics replaces that judgment with a measurement: the correlation coefficient, written r — a single number that captures both the direction of a linear relationship and how tightly the points follow it.
Guess before you learn
A scatter of people's height and shoe size rises tightly, the points close to a straight line. On a scale from −1 to 1, guess the correlation r.
A tight upward cloud sits near +1. Height and shoe size correlate around +0.85 in real data — strong and positive, but not perfect, because people of the same height still vary. Keep your guess; you will calibrate it this folio.
The number r always lands between −1 and 1, and you read it in two halves. Its sign is the direction: positive means the points rise together, negative means one falls as the other climbs. Its size is the strength: the closer the distance of r from 0, toward 1, the more nearly the points lie on one straight line. At exactly 1 or −1 they lie on a line perfectly.
9–12
3–5
r is a closeness score. It runs from −1 to 1. Near 1, two things move up together neatly. Near −1, one goes up as the other goes down. Near 0, they wander apart and follow no straight line.
6–8
The correlation r is one number between −1 and 1. Its sign gives direction: positive rises together, negative moves in opposite directions. Its size gives strength: a value near 1 or −1 means the points hug a straight line, while a value near 0 means no straight-line pattern. Note the catch — r only measures straight-line relationships.
9–12
Correlation r standardizes a relationship. For each point, convert both coordinates to z-scores — how many standard deviations above or below their means each is — then average the products: r = the sum of z_x·z_y divided by (n − 1). When x and y are large together or small together, their z-scores share a sign and the products are positive, pushing r up; when they disagree, the products are negative. Because it uses z-scores, r has no units and does not change if you switch units or swap the axes. It captures only linear association, and a single outlier can move it sharply.
K–2
Two things can grow together. The more you water a plant, the taller it gets. A number close to one says they grow together tightly. A number close to zero says they do not follow each other at all.
Undergrad
r is the sample Pearson coefficient, r = cov(x,y) / (s_x·s_y), invariant under separate linear rescalings of x and y. Its square, r², is the fraction of the variance in one variable accounted for by a straight-line fit to the other. Two cautions stand permanently: r measures linear dependence only — a strong parabola can give r near 0 — and r is not resistant, so bivariate outliers and a restricted range distort it. Anscombe's quartet again: identical r, four different pictures.
Postgrad
The population analogue is ρ = Cov(X,Y) / (σ_X·σ_Y), lying in [−1,1], with the magnitude equal to 1 exactly when Y is an almost-sure affine function of X. The sample r estimates ρ but is biased in small samples; inference typically uses Fisher's variance-stabilizing transform arctanh(r), approximately normal with variance 1/(n − 3). Crucially, ρ = 0 does not imply independence except under joint normality — it nulls only the linear component of dependence. Rank measures (Spearman, Kendall) trade the linearity assumption for monotonicity.
correlation coefficient
A number r between −1 and 1 measuring the direction and the tightness of a linear relationship.
Why is this true?
Why can a strongly curved relationship still have r near 0?
Because r measures only how well a straight line fits. If the points bend up and then down, their above-average and below-average parts cancel in the average of z-score products, so the straight-line score lands near zero even though the pattern is strong and real.
Where does r come from? Turn every coordinate into a z-score — its distance from the mean in standard deviations, the same conversion you used for the normal curve. For each point, multiply its two z-scores. When both are above average, or both below, the product is positive; when they disagree, it is negative. Average those products and you have r.
Compute r from three points' z-scores — the steps fade as you master them
1, 0, 1
1 + 0 + 1 = 2
2 ÷ 2 = 1
r = 1: a perfect positive line
You now have a number for the lean of a cloud: r, from −1 to 1, sign for direction and size for strength. But r only rates a line — it does not draw one, and it cannot predict. For that you need the line of best fit, which is the next folio.
Practice — new ink and old, interleaved
1.Match each z-score to its meaning.
2.A correlation of r = 0.05 between two variables means what?
3.A data value is 80; the mean is 68 and the standard deviation is 8. What is its z-score?
4.A scatter's points fall steadily from upper left to lower right, close to a line. Describe it.
5.Without looking back: what do the sign and the size of r each tell you?
The sign gives the direction — positive rises together, negative moves oppositely — and the size, near 1 versus near 0, gives the strength of the straight-line pattern.
How close were you? Grade yourself honestly — it sets your review date.
6.Computing r divides by each variable's standard deviation. What does standard deviation measure?
7.A scatterplot of hours studied (across) and test score (up) has points rising from lower left to upper right. What is its direction?
8.A z-score of −1.5 tells you the value is:
9.The same right-skewed wait times have a mean of 22 minutes and a median of 14. Which is the more honest headline figure for a typical wait?