--- title: "Correlation coefficient lies between −1 and +1" subject: "Probability and Statistics" chapter: 5 key: "semester_5/probability_and_statistics:correlation-coefficient-between-minus1-plus1" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/correlation-coefficient-between-minus1-plus1 site: BCT Engineering --- # Correlation coefficient lies between −1 and +1 Probability and Statistics · Chapter 5: Correlation and regression - Asked in: 2070 Asar, 4 marks - Video: [Correlation Coefficient lies between -1 and +1](https://www.youtube.com/watch?v=gbZzNhKTl5Q) — RCD Sir - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/correlation-coefficient-between-minus1-plus1 - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-5 ## Question What do you mean by correlation coefficient? Show that correlation coefficient lies between -1 and +1. ## Short answer The Pearson correlation coefficient measures linear association: r = Cov(X,Y)/(σx σy). Proof: let u = (x−x̄)/σx, v = (y−ȳ)/σy; Σu²/n=1, Σv²/n=1. Then Σ(u±v)² ≥ 0 gives 2 ± 2r ≥ 0, hence −1 ≤ r ≤ +1. ## Answer #### Correlation coefficient The (Karl Pearson) correlation coefficient measures the strength and direction of the linear relationship between two variables X and Y: r = Cov(X, Y)/(σx σy) = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)² Σ(y − ȳ)²] r > 0 means variables move in the same direction, r < 0 in opposite directions, r = 0 means no linear relationship. #### Proof that −1 ≤ r ≤ +1 Let u = (x − x̄)/σx and v = (y − ȳ)/σy. Then Σu²/n = 1, Σv²/n = 1 and Σuv/n = r. Since a sum of squares is never negative, Σ(u ± v)² ≥ 0 Σu² + Σv² ± 2Σuv ≥ 0 Dividing by n: 1 + 1 ± 2r ≥ 0 2 ± 2r ≥ 0, i.e. 1 ± r ≥ 0 Taking the upper sign: 1 + r ≥ 0, so r ≥ −1. Taking the lower sign: 1 − r ≥ 0, so r ≤ +1. Hence **−1 ≤ r ≤ +1**. **Equality:** r = +1 when Σ(u − v)² = 0, i.e. u = v for every point (perfect positive linear relation); r = −1 when Σ(u + v)² = 0, i.e. u = −v (perfect negative linear relation). (Alternatively, by the Cauchy–Schwarz inequality, [Σ(x − x̄)(y − ȳ)]² ≤ Σ(x − x̄)² Σ(y − ȳ)², so r² ≤ 1.) ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In the proof, what are u and v? A. u = x − y, v = y − x B. u = (x − x̄)/σx, v = (y − ȳ)/σy C. u = x/σx, v = y/σy D. u = x̄, v = ȳ Answer: B. Standardising gives Σu²/n = 1, Σv²/n = 1 and Σuv/n = r. (Key point: Standardised variables) 2. Which fact is the basis of the proof? A. Σ(u ± v)² ≥ 0, a sum of squares is non-negative B. Σu = 0 C. Σuv = 0 D. Σu² = n² Answer: A. Expanding Σ(u ± v)² ≥ 0 and dividing by n gives 2 ± 2r ≥ 0. (Key point: Key inequality) 3. Which sign gives r ≤ 1? A. Σ(u + v)² ≥ 0 B. Σ(u − v)² ≥ 0 C. Σu² ≥ 0 D. Σuv ≥ 0 Answer: B. Σ(u − v)² ≥ 0 gives 2 − 2r ≥ 0, i.e. r ≤ 1. (Key point: Upper bound) 4. When is r = +1? A. When u = −v for each point B. When Σuv = 0 C. When u = v for every point (perfect positive linear relation) D. When σx = σy Answer: C. r = 1 when Σ(u − v)² = 0, i.e. u = v for all points. (Key point: Equality) ## Related in Probability and Statistics - [Correlation vs regression; marks correlation](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Acorrelation-vs-regression-statistics-maths) — Probability and Statistics, chapter 5 - [22 pairs: means, consistency, correlation](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Asample-22-pairs-xy-correlation) — Probability and Statistics, chapter 5 - [Multiple correlation: expenditure, income, family size](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Amultiple-correlation-food-expenditure) — Probability and Statistics, chapter 5 - [Karl Pearson r and coefficient of determination](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Akarl-pearson-coefficient-determination) — Probability and Statistics, chapter 5 ## Related in other subjects - [Zero reflection coefficient; 3rd order LPF equal termination](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Areflection-zero-third-order-equal) — Filter Design, chapter 5, asked 4× - [Information from reflection coefficient; 3rd order LPF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Areflection-info-third-order-equal) — Filter Design, chapter 5, asked 4× - [Transmission coefficient; 2nd order Butterworth LPF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Atransmission-coefficient-second-order) — Filter Design, chapter 5, asked 3× - [Unity transmission coefficient; 3rd order LPF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aunity-transmission-third-order-equal) — Filter Design, chapter 5, asked 3×