Fundamentals of Business Mathematics and Statistics · Correlation and Regression

Regression Coefficients and Their Properties

Updated 10 October 2026 · Fact-checked

A regression coefficient is the slope of a regression line. byx is the change in y for a unit change in x, and bxy is the reverse. Both have the same sign as r, and r² = byx × bxy. Two regression lines meet at the point (x̄, ȳ), which gives both means.

Understand Regression Coefficients and Their Properties

A regression line predicts one variable from another. The line of y on x predicts y when x is known. Its slope is the regression coefficient byx. The line of x on y predicts x from y. Its slope is bxy.

Both coefficients come from the same data, so they are linked to the correlation coefficient r. The links are byx = r × σy ÷ σx and bxy = r × σx ÷ σy. Multiply them and the standard deviations cancel. This gives r² = byx × bxy. So r is the square root of that product.

The sign of r decides the sign of both coefficients. If the variables move together, r, byx and bxy are all positive. If they move in opposite directions, all three are negative. byx and bxy can never have opposite signs.

The two lines always pass through the point (x̄, ȳ). So if you are given both equations, solving them together gives the two means. This is a favourite exam question.

Correlation and regression are different. Correlation measures the strength and direction of a linear relationship, and r is the same whichever variable you call x. Regression gives an equation for estimating one variable from the other, and the two lines differ. Correlation does not say which variable depends on which. Regression does.

Key formulas to remember

Regression coefficient of y on x: byx = r × σy ÷ σx = Σ(x − x̄)(y − ȳ) ÷ Σ(x − x̄)²; This is the slope of the line of y on x. The denominator is the sum of squared deviations of x. If you divide the top and bottom by n, it becomes covariance ÷ variance of x.
Regression coefficient of x on y: bxy = r × σx ÷ σy = Σ(x − x̄)(y − ȳ) ÷ Σ(y − ȳ)²; This is the slope of the line of x on y. The denominator is the sum of squared deviations of y. If you divide the top and bottom by n, it becomes covariance ÷ variance of y.
Link with correlation: r² = byx × bxy, so r = ±√(byx × bxy); Give r the common sign of byx and bxy.
Sign property: byx, bxy and r always have the same sign; If one is negative, the other is too.
Size property: byx × bxy ≤ 1, and |byx + bxy| ÷ 2 ≥ |r|; The two coefficients always have the same sign. So the AM of their absolute values is at least |r|. This applies to absolute values, because for negative coefficients the plain AM is negative. If the question does not label which line is y on x and a product comes out above 1, you have assigned the equations wrongly, so swap them. If the question does label the lines and the product is above 1, the data are inconsistent. Swapping is not the fix in that case.
Regression lines: y − ȳ = byx (x − x̄) and x − x̄ = bxy (y − ȳ); Both lines pass through (x̄, ȳ).
Slope from an equation ax + by + c = 0: As y on x: byx = −a ÷ b. As x on y: bxy = −b ÷ a; This holds only when the equation is written as ax + by + c = 0, with everything on one side.
Change of origin and scale: If u = (x − a) ÷ h and v = (y − c) ÷ k, then byx = (k ÷ h) × bvu and bxy = (h ÷ k) × buv; Regression coefficients do not change with the origin. They change with the scale.

How to solve Regression Coefficients and Their Properties questions

Use this method when you are given two regression equations and asked for the means, the coefficients or r.

1Write both equations in the form ax + by + c = 0, with all terms on one side. Note the values of a and b for each.
2If the question tells you which line is y on x, use byx = −a ÷ b for that line and bxy = −b ÷ a for the other. If it does not, assume the first line is y on x and the second is x on y.
3Find the product byx × bxy. If it is 1 or less, your assignment is consistent. If it is more than 1, swap the roles of the two equations and recompute.
4Treat this swap test as a check, not a guarantee. If the product is exactly 1, both assignments pass, and the question must say which line is which. A negative product means a sign slip, so recheck.
5Take r = √(byx × bxy). Give it the same sign as byx and bxy.
6For the means, solve the two equations together as simultaneous equations. The values of x and y are x̄ and ȳ.
7If σ values are given, use byx = r × σy ÷ σx or bxy = r × σx ÷ σy to find the missing quantity.
8Check that the sign, the size of r (between −1 and 1) and the means are sensible before marking the option.

Quickest way: Assume, multiply, check

When to use it: Use this for MCQs that give two regression equations and ask for r or the means.

For the means, eliminate one variable between the two equations. Do not rearrange them first.
For r, put both equations in the form ax + by + c = 0. Assume the first is y on x, so byx = −a ÷ b using its own a and b.
Take bxy from the second equation as −b ÷ a, using that equation's own a and b. Multiply the two.
If the product is above 1, swap the assignment. If it is exactly 1, the question must say which is which. Then take the square root.
Look at the signs. If byx and bxy are negative, r is negative.

Common mistakes in Regression Coefficients and Their Properties

Using −a ÷ b as the slope for both equations.
Students apply the y-on-x slope formula to the x-on-y line as well.
Fix: For the line treated as x on y, use bxy = −b ÷ a. Write both equations as ax + by + c = 0 first.
Always giving r a positive sign.
The square root symbol suggests a positive answer.
Fix: Check the sign of byx and bxy. If both are negative, r is negative.
Not checking that byx × bxy is at most 1.
Students assign the equations without testing the assignment.
Fix: If the product is greater than 1, swap the roles of the equations. r² can never exceed 1.
Mixing up σx and σy in byx = r σy ÷ σx.
The two formulas look alike.
Fix: Remember that byx has σx in the denominator because x is the variable you start from.
Finding the means by averaging the constants or by setting x = 0.
Students forget that the means are the point where the two lines meet.
Fix: Solve the two equations simultaneously. The solution is (x̄, ȳ).
Thinking that a change of scale does not affect regression coefficients.
Students mix this up with the origin rule.
Fix: The origin change has no effect. A change of scale multiplies or divides the coefficient by the scale ratio.

Worked examples

Example 1

The two regression lines are 4x − 5y + 33 = 0 and 20x − 9y = 107. Find x̄, ȳ and r. Options for r: (a) 0.36 (b) 0.6 (c) 0.8 (d) 0.45.

Show the solution

Solve for the means. From the first line, 4x = 5y − 33.
Multiply the first equation by 5: 20x − 25y = −165. The second is 20x − 9y = 107.
Subtract: (−25y + 9y) = −165 − 107, so −16y = −272 and ȳ = 17.
Put y = 17 in 4x = 5y − 33: 4x = 85 − 33 = 52, so x̄ = 13.
Assume the first line is y on x: y = (4x + 33) ÷ 5, so byx = 0.8. This matches −a ÷ b = −4 ÷ (−5) = 0.8.
Assume the second is x on y: x = (107 + 9y) ÷ 20, so bxy = 9 ÷ 20 = 0.45. This matches −b ÷ a = 9 ÷ 20.
Product = 0.8 × 0.45 = 0.36, which is less than 1, so the assignment is valid.
r = √0.36 = 0.6. Both coefficients are positive, so r = +0.6.

Answer: x̄ = 13, ȳ = 17 and r = 0.6, option (b).

Example 2

For a data set, byx = −0.4 and bxy = −0.9. Find r. Options: (a) 0.36 (b) −0.36 (c) 0.6 (d) −0.6.

Show the solution

Use r² = byx × bxy.
Product = (−0.4) × (−0.9) = 0.36.
The product is less than 1, so the values are consistent.
r = √0.36 = 0.6 in size.
Both coefficients are negative, so r is negative: r = −0.6.
Options (a) and (b) are r², not r, so they are eliminated.

Answer: r = −0.6, option (d).

Exam tips

Questions on means ask you to solve two equations. Do the elimination directly and avoid rearranging unless you need it.
When asked for r, always check the sign. Examiners put both the positive and negative root among the options.
Eliminate options that are r² instead of r, or have the wrong sign. These are common distractors.
If the product of your coefficients exceeds 1, swap the equations. If the product is exactly 1, the question must tell you which line is which.
Learn the result that the AM of the two coefficients is at least |r| (in absolute values), and r² = byx × bxy. They answer several theory-based MCQs in seconds.

Practice questions from Correlation and Regression

Regression Coefficients and Their Properties in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Regression Coefficients and Their Properties: frequently asked questions

What are the properties of regression coefficients?

Both coefficients have the same sign as r. Their product equals r², so r is the square root of the product. The AM of the absolute values of the two coefficients is at least |r|, and regression coefficients are independent of change of origin but not of scale.

How do I find the mean values from two regression equations?

Both regression lines pass through the point (x̄, ȳ). Solve the two equations as simultaneous equations. The value of x is x̄ and the value of y is ȳ.

What is the relationship between correlation and regression coefficients?

r is the geometric mean of byx and bxy, with the common sign. In symbols, r² = byx × bxy. Also, byx = r σy ÷ σx and bxy = r σx ÷ σy.

What is the difference between correlation and regression?

Correlation measures the strength and direction of a linear relationship, and r is the same for x and y. Regression gives an equation to estimate one variable from another, so there are two different lines. Regression assumes one variable depends on the other.