--- title: "Correlation vs regression; marks correlation" subject: "Probability and Statistics" chapter: 5 key: "semester_5/probability_and_statistics:correlation-vs-regression-statistics-maths" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/correlation-vs-regression-statistics-maths site: BCT Engineering --- # Correlation vs regression; marks correlation Probability and Statistics · Chapter 5: Correlation and regression - Asked in: 2075 Chaitra, 5 marks - Video: [Correlation Vs Regression: Difference Between them with definition & Comparison Chart](https://www.youtube.com/watch?v=Ou2QGSJVd0U) — Key Differences - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/correlation-vs-regression-statistics-maths - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-5 ## Question Differentiate between correlation and regression? From following data find the Karl Pearsons coefficient correlation and interpret the result? | Marks in Statistics | 39 | 65 | 62 | 90 | 82 | 75 | 25 | 98 | 36 | 78 | |---|---|---|---|---|---|---|---|---|---|---| | Marks in Mathematics | 47 | 53 | 58 | 86 | 62 | 68 | 60 | 91 | 51 | 84 | ## Short answer Correlation measures strength and direction of a linear relationship; regression provides an equation to predict one variable from another. For the data, r = 0.780, a strong positive correlation. ## Answer #### Difference between correlation and regression | Basis | Correlation | Regression | |---|---|---| | Purpose | Measures strength and direction of linear relationship | Finds the equation to estimate one variable from another | | Variables | Both treated alike; no dependent/independent distinction | One dependent, one independent | | Symmetry | r(x, y) = r(y, x) | Line of Y on X differs from X on Y | | Value | A pure number between −1 and +1 | Coefficients with units (slope, intercept) | | Cause and effect | Does not imply it | Assumes the independent variable explains the dependent one | | Use | Degree of association | Prediction and estimation | #### Karl Pearson's coefficient r = Sxy / √(Sxx·Syy), where x = marks in Statistics, y = marks in Mathematics, n = 10. Sums (n = 10): ```text x y xy x² y² 39 47 1833 1521 2209 65 53 3445 4225 2809 62 58 3596 3844 3364 90 86 7740 8100 7396 82 62 5084 6724 3844 75 68 5100 5625 4624 25 60 1500 625 3600 98 91 8918 9604 8281 36 51 1836 1296 2601 78 84 6552 6084 7056 --------------------------------------------- 650 660 45604 47648 45784 ``` Σx = 650, Σy = 660, Σxy = 45604, Σx² = 47648, Σy² = 45784 Means: x̄ = 65.0000, ȳ = 66.0000 Sxx = Σx² − (Σx)²/n = 5398.0000 Sxy = Σxy − (Σx)(Σy)/n = 2704.0000 Syy = Σy² − (Σy)²/n = 2224.0000 Slope b = Sxy/Sxx = 2704.0000/5398.0000 = 0.5009 Intercept a = ȳ − bx̄ = 66.0000 − (0.5009)(65.0000) = 33.4398 r = 2704.0000 / √(5398.0000 × 2224.0000) = 2704.0000/3464.8452 = **0.7804** **Interpretation:** r = 0.780 is close to +1, so there is a strong positive linear correlation between marks in Statistics and Mathematics: students scoring high in one tend to score high in the other. About 60.9% (r² = 0.609) of the variation in one subject's marks is explained by the other. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which is a correct difference between correlation and regression? A. Correlation has units; regression is pure B. Correlation distinguishes dependent and independent variables C. Correlation is symmetric (r_xy = r_yx); regression of Y on X differs from X on Y D. Regression is always between -1 and 1 Answer: C. r is symmetric and unit-free; regression lines differ and have units. (Key point: Symmetry difference) 2. Using Sxy = 2704, Sxx = 5398, Syy = 2224, r equals: A. 0.501 B. 0.609 C. 1.216 D. 0.780 Answer: D. r = 2704/√(5398 × 2224) = 2704/3464.85 = 0.7804. (Key point: Value of r) 3. r = 0.78 means: A. Weak negative relationship B. Strong positive linear relationship; high marks in one go with high marks in the other C. No relationship D. 78% of students passed both Answer: B. r close to +1 indicates a strong positive linear correlation. (Key point: Interpretation) 4. What does r² = 0.609 tell us? A. About 60.9% of the variation in one subject's marks is explained by the other B. 60.9% of marks are equal C. r is 0.609 D. 39.1% of students scored high Answer: A. r² is the proportion of variation in one variable explained by the other. 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