Quantitative Aptitude · Correlation and Regression

Regression Coefficients and Their Properties

Updated 1 October 2026 · Fact-checked

Regression coefficients byx and bxy are the slopes of the two regression lines. byx = r·σy/σx and bxy = r·σx/σy. Their product equals r², both carry the sign of r, and the lines meet at the means. To solve questions, assign the lines, check byx·bxy ≤ 1, then find r or the means.

Understand Regression Coefficients and Their Properties

A regression line predicts one variable from the other. The line of y on x predicts y for a given x. The line of x on y predicts x for a given y. These are two different lines, not one.

The slope of each line is a regression coefficient. byx is the slope of y on x: it tells you how much y changes, on average, for a one-unit rise in x. bxy is the slope of x on y: it tells you how much x changes for a one-unit rise in y.

Both slopes are tied to the correlation coefficient r. Each equals r times a ratio of standard deviations. Since standard deviations are always positive, byx, bxy and r always have the same sign. If r is zero, both coefficients are zero.

Multiply the two slopes and the standard deviations cancel. You get r². So r is the geometric mean of byx and bxy, with the common sign. This single link answers most exam questions.

Both regression lines pass through the point (x̄, ȳ). So if you are given the two equations, solving them together gives the means.

Key formulas to remember

Regression coefficient of y on x: byx = r × σy ÷ σx = Cov(x, y) ÷ σx²; Slope of the line y − ȳ = byx (x − x̄). Divide by the variance of x.
Regression coefficient of x on y: bxy = r × σx ÷ σy = Cov(x, y) ÷ σy²; Slope of the line x − x̄ = bxy (y − ȳ). Divide by the variance of y.
Link with correlation: r² = byx × bxy, so r = ± √(byx × bxy); Take the sign of the regression coefficients. Both must have the same sign.
Sign property: byx, bxy and r have the same sign; If one coefficient is positive and the other negative, the data is wrong.
Size property: |byx × bxy| ≤ 1; Since |r| ≤ 1. If one coefficient is numerically more than 1, the other must be less than 1.
AM and r: (byx + bxy) ÷ 2 ≥ r; For positive coefficients, the arithmetic mean of the two is at least r (AM ≥ GM). For negative ones, compare absolute values.
Lines meet at the means: Both lines pass through (x̄, ȳ); Solve the two equations together to get x̄ and ȳ.
Change of origin and scale: If u = (x − a) ÷ c and v = (y − b) ÷ d, then byx = (d ÷ c) × bvu and bxy = (c ÷ d) × buv; Regression coefficients do not change with change of origin. They do change with change of scale.
Angle between the regression lines: tan θ = |(1 − r²) ÷ r| × σxσy ÷ (σx² + σy²); If r = 0 the lines are perpendicular. If r = ±1 the lines coincide (θ = 0).

How to solve Regression Coefficients and Their Properties questions

Use this method for questions that give two regression equations or ask for r, a coefficient, or the angle.

1Write down what is given: equations, standard deviations, r, or means.
2If two equations are given, decide which is y on x and which is x on y. Try one assignment and find byx and bxy.
3Check that byx × bxy ≤ 1 and that both have the same sign. If the product exceeds 1, swap the assignment.
4Find r = ± √(byx × bxy). Use the common sign of the coefficients.
5To find the means, solve the two equations together as simultaneous equations. The answer is (x̄, ȳ).
6If standard deviations are needed, use byx = r σy/σx or bxy = r σx/σy.
7For the angle, put r, σx and σy into the tan θ formula. Use |r| in the bracket and give the acute angle.
8Match your answer to the options and check the sign of r.

Quickest way: Assign, check, then take the root

When to use it: Use this for MCQs with two regression equations where you need r or the means.

For the means, solve the two equations at once. Eliminate one variable and read both values. You can also test each option's means in both equations.
For r, take the coefficient of x in one line when written as y = ... and the coefficient of y in the other when written as x = ... . Ignore the other way round at first.
If the product is more than 1, swap the roles. This is the only assignment that works.
Take the square root of the product. Give it the sign of the coefficients. Options with the wrong sign can be dropped immediately.
If no option has the right sign or the product exceeds 1 under both assignments, skip it and move on. Wrong answers cost 0.25 marks.

Common mistakes in Regression Coefficients and Their Properties

Writing byx = r σx/σy, with the standard deviations swapped
Both formulas look alike and students learn them without a rule.
Fix: Remember that byx has σy on top because y is the variable being predicted. The first letter of the subscript is the top one: byx has y over x.
Taking r as positive when both regression coefficients are negative
Students take the square root and forget the sign.
Fix: The square root only gives the size. Give r the same sign as byx and bxy. Both negative means r is negative.
Assigning the two given equations to y on x and x on y without checking
Students assume the first equation is y on x.
Fix: Try an assignment and check byx × bxy ≤ 1. If it exceeds 1, swap the equations.
Reading the coefficient directly from an equation not in the right form
In 3x + 2y = 26, students take 3 as the slope.
Fix: Rewrite the equation. For y on x write y = ... and read the coefficient of x. For x on y write x = ... and read the coefficient of y.
Saying regression coefficients are unchanged by a change of scale
Students mix this up with the fact that r does not change with origin or scale.
Fix: Regression coefficients are unchanged by origin shifts only. A scale change multiplies them by the ratio d/c or c/d. r is unchanged apart from its sign if the scale factors have opposite signs.
Using the means as the intercepts or forgetting that the lines intersect at the means
Students try to find the means by setting x or y to zero.
Fix: Solve the two equations simultaneously. The intersection point is (x̄, ȳ).

Worked examples

Example 1

The regression coefficients of two variables are byx = 0.8 and bxy = 0.45. The correlation coefficient r is: (A) 0.36 (B) 0.60 (C) 0.625 (D) 0.90

Show the solution

Both coefficients are positive, so r is positive.
r² = byx × bxy = 0.8 × 0.45 = 0.36.
r = √0.36 = 0.6.
Check: the product 0.36 is at most 1, so the data is valid.

Answer: (B) 0.60

Example 2

The two regression lines are 3x + 2y = 26 and 6x + y = 31. The means x̄ and ȳ and the correlation coefficient r are: (A) x̄ = 4, ȳ = 7, r = 0.5 (B) x̄ = 7, ȳ = 4, r = −0.5 (C) x̄ = 4, ȳ = 7, r = −0.5 (D) x̄ = 4, ȳ = 7, r = −0.25

Show the solution

Means: from 6x + y = 31, y = 31 − 6x.
Put this in 3x + 2y = 26: 3x + 62 − 12x = 26, so −9x = −36 and x = 4.
Then y = 31 − 24 = 7. So x̄ = 4 and ȳ = 7. Check in the first line: 12 + 14 = 26.
Try 3x + 2y = 26 as y on x: y = 13 − 1.5x, so byx = −1.5.
Try 6x + y = 31 as x on y: x = (31 − y) ÷ 6, so bxy = −1/6.
Product = 1.5 × 1/6 = 0.25, which is at most 1, so this assignment is valid.
r² = 0.25, so |r| = 0.5. Both coefficients are negative, so r = −0.5.

Answer: (C) x̄ = 4, ȳ = 7, r = −0.5

Example 3

For two variables, r = 0.6, σx = 3 and σy = 4. The value of tan θ, where θ is the acute angle between the two regression lines, is: (A) 64/125 (B) 16/15 (C) 12/25 (D) 4/5

Show the solution

Formula: tan θ = |(1 − r²) ÷ r| × σxσy ÷ (σx² + σy²).
1 − r² = 1 − 0.36 = 0.64.
(1 − r²) ÷ r = 0.64 ÷ 0.6 = 16/15.
σxσy = 12 and σx² + σy² = 9 + 16 = 25, so the ratio is 12/25.
tan θ = 16/15 × 12/25 = 192/375 = 64/125.

Answer: (A) 64/125

Exam tips

Questions usually give two regression equations and ask for r, the means, or one coefficient. Practise the assign, check, root routine until it takes under a minute.
Always check the sign of r before you pick an option. Options often differ only in sign.
Memorise r² = byx × bxy and the means shortcut. Many MCQs can be done with just these two.
Property questions ask which statement is true. Remember: same sign, product at most 1, AM at least r, unchanged by origin shift but not by scale.
For the angle formula, remember the special cases. If r = 0 the lines are perpendicular, and if r = ±1 the lines coincide.

Practice questions from Correlation and Regression

Regression Coefficients and Their Properties: frequently asked questions

What is the difference between byx and bxy?

byx is the slope of the regression line of y on x and equals r σy/σx. bxy is the slope of the line of x on y and equals r σx/σy. They are equal only when σx = σy.

How do I find r from two regression equations?

Assign the lines so that byx × bxy is at most 1. Then r = ±√(byx × bxy), with the sign shared by both coefficients. The product of the two slopes gives r² directly.

How do I find the mean of x and y from two regression lines?

Both lines pass through (x̄, ȳ). Solve the two equations as simultaneous equations. The values you get for x and y are the means.

Can both regression coefficients be greater than 1?

No. Their product equals r², which cannot exceed 1. So if one is numerically greater than 1, the other must be numerically less than 1.

Does a change of scale affect regression coefficients?

Yes. If x and y are divided or multiplied by constants, the coefficients are multiplied by the ratio of those constants. A change of origin alone does not change them.