Association & Correlation

A correlation only says the two series move up and down together in the observed data. It cannot say ice cream causes drownings: something else, such as hot weather, can drive both, and correlation carries no direction while causation does.

Sample covariance measures whether two variables move together, but it carries the product of their units, so its size means little alone. Pearson's r divides covariance by both standard deviations, giving a unitless number between -1 and 1.

Regression to the mean is the tendency for an extreme measurement to be followed by a less extreme one on remeasurement. Expect it whenever two measurements are correlated but not perfectly, because chance helped produce the extreme.

Spearman's rho is a correlation computed on the ranks of the data instead of the raw values. It measures how well two variables move together in a consistently increasing or decreasing pattern, not just along a straight line.

Cramer's V equals sqrt(chi2 / (n x (min(r, c) - 1))), with n the total count and r, c the table dimensions. It rescales the chi-square statistic onto 0 to 1, with no direction.

Anscombe's quartet is four eleven-point datasets sharing the same means, variances, correlation near 0.82 and fitted line to two decimals, yet their scatterplots look completely different. Summary statistics do not identify a dataset - plot before concluding.

Squaring Pearson's r gives shared variance: 0.64 means 64 percent of the variation in exam scores moves with variation in study hours, leaving 36 percent unaccounted for. It is a proportion of variance, not of scores.

About 1.2 standard deviations above the mean. In standard-deviation units the expected second score is the correlation times the first score, so 0.6 times 2 gives 1.2. The prediction is pulled toward the mean because the correlation is below 1.

Kendall's tau compares every pair of observations. A pair is concordant when both variables order it the same way, discordant when they disagree. Tau is concordant minus discordant pairs, divided by the total number of pairs.

Point-biserial r is just Pearson's correlation with the binary variable coded 0 and 1. It rescales the gap between the two group means, and shrinks as the two groups become more unequal in size.

Pearson's r measures straight-line association only. On a parabola centred at zero, the rising and falling halves contribute deviation products of opposite sign that cancel exactly, so r is zero even though y is perfectly determined by x.

The diagonal holds each variable's sample variance, because a variable's covariance with itself is its variance. Off-diagonal entries hold the pairwise sample covariances, and the matrix is symmetric since covariance does not depend on pair order.

The relationship is perfectly monotone but strongly curved. Spearman's rho sees only the ordering, which matches exactly, so it hits 1. Pearson's r measures closeness to a straight line, and an exponential curve is far from straight.

Use the correlation ratio eta. Eta-squared is the between-group sum of squares over the total sum of squares: the share of salary variance accounted for by department membership. It runs 0 to 1.

Admission selects on the score, so admitted students span a narrow score range. Correlation scales with how much the predictor varies relative to the leftover scatter, so truncating that spread shrinks r even though the underlying relationship is unchanged.

Pearson's r sums cross-products of deviations, so one point far from both means can contribute a term that outweighs all the others and dominates the statistic. Plot the scatter and recompute r with each row removed to catch it.

Some of the drop is regression to the mean: the junctions were chosen for having an extreme year. Estimate the camera effect against untreated junctions selected by the same rule, not against the treated sites' own worst year.

Ties break the simple formulas. Spearman's rho needs averaged midranks for each tied block and a full correlation on them. Kendall's tau needs the tie-corrected tau-b, because tau-a cannot reach 1 once many pairs are tied.

Change the evaluation, not the targeting. Selecting the worst segment guarantees a rebound whatever the programme does, so require an untreated comparison chosen by the same extreme rule and a published expected-rebound figure any claimed effect must beat.

Regression to the mean, not the praise. Exceptional landings are partly luck, so the next attempt is usually closer to the trainee's normal standard whatever the instructor says. The same logic makes criticism after a terrible landing look effective.

Check sample size and table shape first: Cramer's V is biased upward, so a large sparse table on few rows gives sizeable values from noise. Then read the cell counts and conditional row proportions.

A state-level correlation describes states, not people. Averaging discards within-state variation, so the state-level coefficient can be stronger than, or opposite in sign to, the individual-level one. Reading it as a fact about voters is the ecological fallacy.

Pearson's r is standardised, so it reports how tightly points hug a straight line, not how steep that line is. Dividing by both spreads removes the units, leaving r free to accompany any slope at all.

They are different scales, not competing estimates of one quantity. Tau is a difference of pair-agreement probabilities; Spearman's rho is a correlation of rank positions that weights large rank displacements heavily. Tau typically lands lower.