---
title: "Regression coefficients when equal; properties"
subject: "Probability and Statistics"
chapter: 5
key: "semester_5/probability_and_statistics:regression-coefficients-equal-three-properties"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/regression-coefficients-equal-three-properties
site: BCT Engineering
---

# Regression coefficients when equal; properties

Probability and Statistics · Chapter 5: Correlation and regression

- Asked in: 2074 Asoj, 5 marks
- Video: [Properties Of Regression](https://www.youtube.com/watch?v=ZFR5VJ1t8wI) — Dr. Viney Narrang Chugh
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/regression-coefficients-equal-three-properties
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-5

## Question

What are the two regression coefficients and what do they represent when these two will be same? Write any three properties of regression coefficient.

## Short answer

The two regression coefficients are slopes b_yx and b_xy, giving average change in Y for X and X for Y. They are equal when the standard deviations are equal, and then both equal r. Properties: r is their geometric mean, both have the same sign as r, and their arithmetic mean is at least |r|.

## Answer

#### Regression coefficients

The two regression coefficients are the slopes of the regression lines:

- **b_yx = r·σy/σx**: regression coefficient of Y on X. It gives the average change in Y for a unit change in X.
- **b_xy = r·σx/σy**: regression coefficient of X on Y. It gives the average change in X for a unit change in Y.

#### When the two coefficients are equal

If b_yx = b_xy, then r·σy/σx = r·σx/σy. For r ≠ 0 this gives σx = σy. Then

b_yx = b_xy = r

So the two coefficients are equal when both variables have the same standard deviation, and each then equals the correlation coefficient. (If r = 0, both coefficients are 0 and are trivially equal.) Since b_yx·b_xy = r², equal coefficients also give r² = b², i.e. r = b, because b has the same sign as r.

#### Three properties

1. r = ±√(b_yx · b_xy): the correlation coefficient is the geometric mean of the two regression coefficients, with the same sign as both.
2. Both coefficients have the same sign as r, and they cannot both exceed 1 in magnitude (b_yx·b_xy = r² ≤ 1).
3. The arithmetic mean of the coefficients is at least |r| in magnitude: |b_yx + b_xy|/2 ≥ |r| (for positive r, (b_yx + b_xy)/2 ≥ r). Also, regression coefficients are independent of a change of origin but depend on a change of scale.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does the regression coefficient b_yx = r·σy/σx represent?
   A. Average change in X for a unit change in Y
   B. Average change in Y for a unit change in X
   C. The correlation between Y and X
   D. The ratio of the means of Y and X
   Answer: B. b_yx is the slope of the regression line of Y on X, so it gives the average change in Y per unit change in X. (Key point: Meaning of b_yx)

2. If b_yx = b_xy and r ≠ 0, what must be true?
   A. r = 0
   B. Means of X and Y are equal
   C. Standard deviations of X and Y are equal
   D. Variances differ but r = 1
   Answer: C. r·σy/σx = r·σx/σy gives σx = σy, and then each coefficient equals r. (Key point: When coefficients are equal)

3. Which relation between r and the regression coefficients is correct?
   A. r = b_yx + b_xy
   B. r = ±√(b_yx · b_xy), same sign as the coefficients
   C. r = b_yx / b_xy
   D. r = (b_yx − b_xy)/2
   Answer: B. r is the geometric mean of the two regression coefficients and has their sign. (Key point: Relation with r)

4. Regression coefficients are affected by which change?
   A. Change of origin only
   B. Neither origin nor scale
   C. Both origin and scale
   D. Change of scale only
   Answer: D. They are independent of change of origin but depend on change of scale. (Key point: Change of origin and scale)

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