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Class 11 Correlation: cheat sheet
CBSE Social Science. How two variables move together: scatter diagrams, positive and negative correlation, Karl Pearson’s coefficient r and Spearman’s rank correlation.
Key ideas
- Correlation measures the direction and intensity of a linear relationship between two variables.
- Positive: both move the same way (income and consumption); negative: one rises as the other falls (price and quantity demanded).
- A scatter diagram shows the pattern at a glance: points rising, falling, on a line, or scattered with no pattern.
- r lies between −1 and +1, has no unit, and does not change if a constant is added or every value is scaled.
- r = 0 means no linear relationship; a curved relation may still exist.
- Spearman’s method uses ranks, so it suits qualities such as beauty or honesty and data with extreme values.
- Correlation does not prove cause and effect: it may be spurious or due to a third factor.
Remember
| Pearson’s r (deviations from the means) | r = Σxy ÷ √(Σx² × Σy²) |
| From covariance | r = covariance ÷ (σx × σy) |
| Spearman’s rank correlation | rs = 1 − 6Σd² ÷ (n³ − n), d = difference in ranks |
| Range of r | −1 ≤ r ≤ +1 |
| Reading r | +1 perfect positive; −1 perfect negative; near ±0.9 high; near ±0.1 low |
Watch out
- Adding the rank differences instead of their squares for Σd² (or forgetting that the plain differences add to zero).
- Using n² − 1 or n² in Spearman’s formula instead of n³ − n.
- Accepting a value of r above 1 or below −1: it means a calculation slip.
- Reading a high r as proof that one variable causes the other.
Worked example
Find Karl Pearson’s coefficient of correlation for X: 17, 6, 17, 9, 6 and Y: 94, 61, 95, 67, 63 (to two decimal places).
Working: Σxy = 380, Σx² = 126, Σy² = 1160; r = Σxy ÷ √(Σx² × Σy²)
Answer: 0.99
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