- 2075 Chaitra · 5 marks
Correlation vs regression; marks correlation
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.
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 |
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):
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.
✓ Answer checked by Claude Sonnet 5.5
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