Scatter diagrams
- A scatter diagram plots bivariate data, with the explanatory variable on the horizontal axis.
- Positive correlation: as one variable increases, the other increases. Negative correlation: as one increases, the other decreases. No correlation: no pattern.
- Correlation can be strong (points close to a line) or weak (points scattered).
Correlation isn't causation
- Two variables can be correlated without one causing the other. Ice cream sales and drownings both rise in summer: a third (lurking) variable, temperature, affects both.
- Only a well-designed experiment can show that one variable causes a change in another.
Spearman's rank correlation coefficient
- Rank each set of data, find the difference in ranks (d) for each item, then use rs = 1 − 6Σd² ÷ (n(n² − 1)).
- rs is between −1 and +1. +1 means the rankings agree perfectly, −1 means they are exactly opposite, and 0 means no agreement.
- Example: 5 items with Σd² = 4. 6 × 4 = 24 and 5 × 24 = 120, so rs = 1 − 24 ÷ 120 = 0.8.
Pearson's product moment correlation coefficient
- The PMCC (r) measures the strength of linear correlation, also from −1 to +1. You interpret it rather than calculate it by hand: 0.9 is strong positive, −0.3 is weak negative.
- Spearman's is used for ranked data or when the relationship isn't a straight line; the PMCC is used for linear relationships.
Key terms
- Scatter diagram
- A graph plotting pairs of values for bivariate data.
- Positive correlation
- As one variable increases, the other increases.
- Negative correlation
- As one variable increases, the other decreases.
- Causation
- When a change in one variable directly causes a change in another.
- Spearman's rank correlation coefficient
- A measure of agreement between two sets of rankings.
- PMCC
- A measure of the strength of linear correlation.
- Lurking variable
- A third variable that affects both variables being studied.