Correlation and Regression for CA Foundation
Correlation and regression: scatter diagrams, Karl Pearson's and Spearman's coefficients, regression lines and their properties.
Quantitative AptitudePaper 3, Statistics11 MCQs2 lessons2 flashcards3 memory tricks
Practise Correlation and Regression free
What you'll learn
- Correlation
Measure and interpret correlation. - Regression
Use regression lines and coefficients.
Key formulas
| Pearson's r | Cov(x, y) / (σₓ σᵧ) |
|---|---|
| Spearman's R | 1 − 6Σd² / [n(n² − 1)] |
| Range | −1 ≤ r ≤ +1 |
| Coefficient of determination | r² |
| Regression of y on x | y − ȳ = bᵧₓ (x − x̄) |
| Regression of x on y | x − x̄ = bₓᵧ (y − ȳ) |
| Correlation from coefficients | r = ±√(bᵧₓ × bₓᵧ) |
| bᵧₓ | r σᵧ / σₓ |
Concepts explained
Correlation coefficient
Karl Pearson's r measures the strength and direction of linear relationship, from −1 to +1.
What it means
- r = Cov(x, y)/(σₓσᵧ).
- +1 perfect positive, −1 perfect negative, 0 no linear correlation.
- Independent of change of origin and scale.
- r² (coefficient of determination) = share of variation explained.
- r = 0.8
Answer: r² = 0.64
- Price and demand
Answer: Negative correlation
Related terms
- Covariance: average product of deviations
- Spurious correlation: correlation without causation
Exam tip: Correlation shows association, not cause and effect.
Spearman's rank correlation
For ranked data, R = 1 − 6Σd²/[n(n² − 1)], where d is the difference in ranks.
What it means
- Used when data are ranks or qualitative (beauty, honesty).
- Σd always equals 0 (a check).
- For tied ranks, give the average rank and add a correction.
- Range −1 to +1, like r.
- n = 5, Σd² = 20
Answer: R = 0
- n = 5, Σd² = 10
Answer: R = 0.5
Related terms
- Tied ranks: average rank and correction factor
- Judges' ranking: classic application
Exam tip: n(n² − 1): for n = 5 it's 120; for n = 10 it's 990.
Regression lines and coefficients
Regression lines estimate one variable from another; both pass through the means, and r is the geometric mean of the two regression coefficients.
What it means
- y on x: y − ȳ = bᵧₓ(x − x̄); x on y: x − x̄ = bₓᵧ(y − ȳ).
- r = ±√(bᵧₓ bₓᵧ), with the coefficients' common sign.
- Lines intersect at (x̄, ȳ); coincide if r = ±1; perpendicular if r = 0.
- Coefficients depend on scale but not origin.
- 0.8 and 0.45
Answer: r = 0.6
- 2x + 3y = 26; 6x + y = 30
Answer: x̄ = 4, ȳ = 6
Related terms
- Least squares: method used to fit regression lines
- Standard error of estimate: accuracy of predictions
Exam tip: If both b's are above 1, something is wrong: their product (r²) can't exceed 1.
Memory tricks
R = 1 − 6Σd²/[n(n² − 1)].
r = ±1 lines coincide; r = 0 perpendicular; intersect at (x̄, ȳ).
r = ±√(bᵧₓ × bₓᵧ).
Try 3 CA Foundation-style questions
Q1. Which pair is likely to show positive correlation?
- Price and quantity demanded
- Speed of a car and time taken for a trip
- Height and weight of adults
- Hours of TV watched and exam marks
Show answer
Answer: (3) Height and weight of adults
Taller people tend to weigh more: both move in the same direction.
Q2. Karl Pearson's coefficient of correlation always lies between
- −1 and +1
- 0 and 1
- −∞ and +∞
- 0 and 100
Show answer
Answer: (1) −1 and +1
−1 is perfect negative correlation, +1 perfect positive, 0 no linear correlation.
Q3. If the regression coefficients are bᵧₓ = 0.8 and bₓᵧ = 0.45, the correlation coefficient is
- 0.36
- 0.6
- 1.25
- 0.8
Show answer
Answer: (2) 0.6
r = √(bᵧₓ × bₓᵧ) = √0.36 = 0.6, positive because both coefficients are positive.
Practise all 11 questions Free · explanations for every option · see where you go wrong