Quick revision sheet for Chapter 6: Correlation (Statistics for Economics). For diagrams, worked examples and full explanations, read the complete notes.
1 Key definitions
Learn these
CorrelationStudies and measures the direction and intensity of relationship among variables; measures covariation, not causation
Positive correlationVariables move in the same direction: income and consumption; temperature and ice-cream sales
Negative correlationVariables move in opposite directions: price of apples and demand; rate of interest and demand for funds
Linear relationOne that can be represented by a straight line; a curve means non-linear
Scatter diagramValues of two variables plotted as points; visual, no number, shows any type of relation
Karl Pearson’s coefficient (r)Numerical measure of linear relationship; also product moment correlation coefficient, simple correlation coefficient
CovarianceCov(X, Y) = Σxy/N; its sign is the sign of r
Spearman’s rank correlationBased on ranks instead of values; British psychologist C.E. Spearman
AttributesVariables that cannot be measured numerically: intelligence, physical appearance, honesty
Correction factor(m³ − m)/12 for repeated ranks, one for every group of equal values
2 Three things that produce correlation
| Source | Book’s example |
|---|---|
| Cause and effect | Price and quantity demanded; low rainfall and low agricultural productivity |
| Simple coincidence | Migratory birds and birth rates; shoe size and money in your pocket |
| A third variable | Ice-cream sales and deaths by drowning, temperature behind both |
3 The seven scatter patterns
| Figure | Points | Relation |
|---|---|---|
| 6.1 | Around a rising line | Positive |
| 6.2 | Around a falling line | Negative |
| 6.3 | Scattered | No correlation |
| 6.4 / 6.5 | On a rising / falling line | Perfect positive (+1) / perfect negative (−1) |
| 6.6 / 6.7 | Around a rising / falling curve | Positive / negative non-linear |
4 All formulas
Formulas
$$r = \frac{\sum xy}{N\sigma_x\sigma_y} = \frac{\sum xy}{\sqrt{\sum x^2}\sqrt{\sum y^2}}, \quad x = X – \bar{X},\ y = Y – \bar{Y}$$
$$r = \frac{N\sum XY – (\sum X)(\sum Y)}{\sqrt{N\sum X^2 – (\sum X)^2}\;\sqrt{N\sum Y^2 – (\sum Y)^2}}$$
$$U = \frac{X – A}{h},\ V = \frac{Y – B}{k} \Rightarrow r_{XY} = r_{UV}$$
$$r_s = 1 – \frac{6\sum D^2}{n^3 – n}, \qquad r_s = 1 – \frac{6\left[\sum D^2 + \sum \frac{m^3 – m}{12}\right]}{n(n^2 – 1)}$$
Checks: Σx = 0, Σy = 0, ΣD = 0. Corrections: m = 2 → 0.5, m = 3 → 2, m = 4 → 5.
5 Properties of r
- No unit; a pure number
- $-1 \le r \le 1$; outside means an error in calculation
- Unaffected by change of origin and scale (h, k of the same sign)
- r = 0: no linear relation, a non-linear one is possible (not independence)
- r = ±1: perfect, exact linear relation; close to ±1 strong, close to 0 weak
- Use only for linear relations; check the scatter diagram first
6 Pearson versus Spearman
| Basis | Pearson (r) | Spearman (rs) |
|---|---|---|
| Based on | Actual values | Ranks |
| When | Precisely measured data | Qualitative variables, no measuring tools, non-linear but clear direction |
| Extreme values | Affected | Not affected |
| Accuracy | Higher | Generally lower, $r_k \le r$ (equal if first differences are constant) |
7 Answers to the book’s examples
| Example | Topic | Answer |
|---|---|---|
| 1 | Farmers’ schooling and yield per acre (Σxy = 42, Σx² = 112, Σy² = 38) | r = 0.644 |
| Activity table | National income growth and Gross Domestic Saving | r = 0.32 |
| 2 | Price index and money supply (step deviation) | r = 0.987 ≈ 0.99 |
| 3 | Three judges (n = 5) | A-B 0.3, A-C 0.5, B-C −0.4; A and C closest |
| 4 | Marks in Statistics and Economics (ΣD² = 2) | rs = 0.9 |
| 5 | Repeated ranks (ΣD² = 198, correction 3, n = 12) | rs ≈ 0.30 |
8 Repeated ranks: three rules
- Give equal values the mean of the ranks they share: 9th, 10th, 11th → 10
- Give the next value the next rank: 12
- Add (m³ − m)/12 for every group of equal values to ΣD², then multiply by 6
In one line
- Correlation shows direction and intensity, not cause; the scatter diagram sees the shape, r measures linear relation, rs measures the relation between ranks
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