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Correlation & Regression

Paper-II Statistics (Post-specific) · Statistics Syllabus

Useful for: SSC CGL

Overview

Correlation measures the strength and direction of a linear relationship; regression gives the line to predict one variable from another.

Key points

Pearson’s r lies between −1 and +1.
Spearman’s rank correlation ρ = 1 − 6Σd²/[n(n² − 1)].
Regression coefficient b_yx = r × σy/σx; b_xy = r × σx/σy.
r = ±√(b_yx × b_xy), with the sign of the regression coefficients.
Both regression lines pass through the point of means (x̄, ȳ).

Explanation

Both regression coefficients have the same sign, and their product cannot exceed 1.
Correlation does not imply causation.
If r = 0, the regression lines are perpendicular; if r = ±1, they coincide.
r is unaffected by change of origin and scale (sign may change if scale is negative).

Step by step

Identify the measure or test asked and its formula.
Organise the data (table, totals) before computing.
Interpret the result (sign, size, significance).

Examples

b_yx = 0.8, b_xy = 0.45 → r = √0.36 = 0.6.
n = 5, Σd² = 4 → ρ = 1 − 24/(5 × 24) = 0.8.
Regression lines intersect at the means, e.g. (x̄, ȳ) = (10, 20).

Traps and advanced points

Trap: Taking r as the average of the regression coefficients.
Trap: Mixing b_yx and b_xy.
Trap: Reading causation from correlation.

Quick revision

r = √(b_yx b_xy).
Spearman 1 − 6Σd²/n(n²−1).
Lines meet at the means.

Practice MCQs

The correlation coefficient r always lies between
1. −∞ and +∞
2. 0 and 1
3. −1 and +1
4. 0 and 100
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Answer: (c) −1 and +1. Values at the ends show perfect correlation.
r = 0 indicates
1. perfect positive correlation
2. no linear correlation
3. perfect negative correlation
4. a strong relationship
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Answer: (b) no linear correlation. There may still be a non-linear relationship.
The product of the two regression coefficients equals
1. 1
2. r
3. r²
4. 0
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Answer: (c) r². b_yx × b_xy = r².
The two regression lines intersect at
1. (0, ȳ)
2. the origin
3. (x̄, ȳ)
4. (x̄, 0)
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Answer: (c) (x̄, ȳ). Both lines pass through the means.
Spearman's rank correlation is used when
1. data are given as ranks
2. data are only nominal
3. there is one variable
4. the sample size is infinite
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Answer: (a) data are given as ranks. It works on ranked data.
If b_yx = 0.8 and b_xy = 0.5, the correlation coefficient is
1. 1.3
2. 0.4
3. 0.63
4. 0.8
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Answer: (c) 0.63. r = √(0.8 × 0.5) = √0.4 ≈ 0.63, positive as both are positive.

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