Correlation: direction and strength of association

Correlation summarises how variables move together. Pearson’s r lies between −1 and +1 for properly defined nonconstant numerical variables. Positive values mean positive linear association; negative values mean negative linear association; r near 0 means weak linear association, not necessarily no relationship. Correlation does not establish causation—income and examination access may be related because of confounding factors.

Pearson correlation and covariance

A useful expression is r = Cov(X,Y) / (σₓσᵧ) when the conventions for covariance and SD are consistent. The sign depends on whether deviations tend to move together. If one variable is multiplied by a positive constant, Pearson r does not change; multiplication by a negative constant flips its sign. Swapping X and Y does not change r.

Spearman rank correlation

For ranks without ties, ρ = 1 − 6Σd²/[n(n²−1)]. With ties, apply an appropriate tie correction or compute Pearson correlation on the average ranks. Spearman is useful for ordinal data and monotonic relationships; it is not the same as saying every ranked relationship must be perfectly linear.

Worked example: perfect inverse ranks

Four candidates have ranks in Mathematics 1,2,3,4 and ranks in Statistics 4,3,2,1. Rank differences are −3,−1,1,3 and Σd²=20. Spearman ρ = 1 − 6×20/[4×(16−1)] = 1 − 2 = −1.

Regression: prediction, not causal proof

Simple linear regression models a conditional relationship such as Y = a + bX + error. The least-squares slope for predicting Y from X is bᵧₓ = Cov(X,Y)/Var(X), assuming Var(X)>0. The regression of X on Y is a different line unless correlation is perfect. Regression coefficients can exceed 1 in magnitude because they have units, unlike Pearson r. Predictions outside observed X values require special caution.

Index numbers

Index numbers compare a chosen measure across time or place relative to a base usually normalised to 100. The simple price relative = (current price/base price) × 100. Common weighted price indices include Laspeyres using base-period quantities and Paasche using current-period quantities. Fisher’s ideal price index is their geometric mean under standard definitions. Consumer price indices weight a basket according to their underlying methodology; using the index change to infer all individual prices is incorrect.

Worked example: index interpretation

If an index changes from 125 to 135, the rate of increase over that period is (135−125)/125 ×100 = 8%, not 10% and not 135%. If a different base is used, convert or compare correctly; raw levels from unrelated base years are not immediately comparable.

Sampling: how we obtain representative information

Sampling error arises from observing a sample rather than census; non-sampling error includes nonresponse, measurement, coverage and processing issues. Increasing sample size helps some random errors but does not fix a biased sampling frame.

Practice MCQs

Q1. Pearson r = −0.9 indicates (A) strong negative linear association (B) 90% proven causation (C) no association (D) positive slope necessarily. Answer: A. Magnitude measures linear strength, not causality.

Q2. Choosing every 10th employee after a random start is (A) cluster (B) systematic (C) purposive (D) census. Answer: B. Regular interval defines systematic sampling.

Q3. An index rises from 120 to 126. Percentage increase = (A) 6% (B) 5% (C) 26% (D) 120%. Answer: B. Six divided by 120 equals 5%.

FAQs

Does zero Pearson correlation mean independence?

No, not in general. Nonlinear dependence may remain.

Is a regression coefficient constrained between −1 and +1?

No. That restriction applies to the correlation coefficient, not an unstandardised slope.

Do large samples remove selection bias?

No. A large but systematically unrepresentative sample may remain biased.