Strategic Cost Management · Business Forecasting Models - Time Series and Regression Analysis

Simple Linear Regression Analysis: Regression Lines and Coefficients

Updated 11 October 2026 · Fact-checked

Simple linear regression fits a straight line that estimates one variable from another. The line of Y on X is Y − Ȳ = b_yx (X − X̄), and the line of X on Y is X − X̄ = b_xy (Y − Ȳ). Find the means and coefficients, write the line, then substitute the given value.

Understand Simple Linear Regression Analysis

Regression is a way to estimate the average value of one variable when you know the value of another. In costing, you may estimate total overhead from machine hours, or sales from advertising spend. The variable you estimate is the dependent variable. The variable you use to estimate it is the independent variable.

In simple linear regression there is one independent variable and the relationship is a straight line. The line is fitted by the method of least squares. It makes the sum of squared vertical gaps between the actual points and the line as small as possible.

There are two regression lines. The line of Y on X estimates Y from X. The line of X on Y estimates X from Y. They are different lines because each one minimises errors in a different direction. They pass through the same point (X̄, Ȳ), the means. They coincide only when correlation is perfect (r = +1 or −1).

The regression coefficient is the slope of a line. b_yx is the average change in Y for a one-unit change in X. b_xy is the average change in X for a one-unit change in Y. Both always carry the same sign as the correlation coefficient r.

Correlation and regression are related but not the same. Correlation measures the strength and direction of a linear relationship, and it is symmetric: r between X and Y equals r between Y and X. Regression gives an equation for estimation and is not symmetric, as it treats one variable as dependent. Correlation has no units and lies between −1 and +1. Regression coefficients carry units and can take any value, subject to the properties below.

Key rules to remember

Regression line of Y on X: Y − Ȳ = b_yx (X − X̄); Use it to estimate Y for a given X.
Regression line of X on Y: X − X̄ = b_xy (Y − Ȳ); Use it to estimate X for a given Y.
Regression coefficient using r and standard deviations: b_yx = r × σy ÷ σx ; b_xy = r × σx ÷ σy; The standard deviation of the dependent variable is in the numerator.
Regression coefficient from raw data: b_yx = [nΣXY − ΣX ΣY] ÷ [nΣX² − (ΣX)²] ; b_xy = [nΣXY − ΣX ΣY] ÷ [nΣY² − (ΣY)²]; Use when actual values are given. Take care with the denominator for each line.
Regression coefficient from deviations: b_yx = Σxy ÷ Σx² ; b_xy = Σxy ÷ Σy² (x = X − X̄, y = Y − Ȳ); Useful when deviations from actual means are small or already given.
Assumed mean method: b_yx = [nΣdxdy − Σdx Σdy] ÷ [nΣdx² − (Σdx)²]; dx = X − A, dy = Y − B. The coefficient is unchanged by shifting the origin, not by changing scale.
Correlation from the coefficients: r² = b_yx × b_xy , so r = ±√(b_yx × b_xy); r takes the common sign of both coefficients.
Properties of regression coefficients: Arithmetic mean of b_yx and b_xy ≥ r ; b_yx and b_xy have the same sign ; b_yx × b_xy ≤ 1; The AM statement holds when r is positive, comparing with its absolute value in general. Both coefficients cannot be greater than 1 in absolute terms.
Slope and intercept form: Y = a + bX, where b = b_yx and a = Ȳ − b X̄; The intercept a is the value of Y when X = 0.

How to solve Simple Linear Regression Analysis questions

Use this method for any question on regression lines, coefficients or estimation.

1Identify which variable is to be estimated. That is the dependent variable. Decide which line you need.
2List what is given: means, standard deviations, r, raw data, or two equations.
3Find the means X̄ and Ȳ. If the data are raw, compute n, ΣX, ΣY, ΣXY, ΣX² and ΣY².
4Compute the regression coefficient needed, using r and the standard deviations or the raw-data formula.
5Write the regression line through the means, for example Y − Ȳ = b_yx (X − X̄), and simplify to Y = a + bX.
6Substitute the given value of the independent variable to get the estimate. State the units.
7If two equations are given, decide which is which, check that b_yx × b_xy ≤ 1, then find r and the means.
8Check the sign of the coefficients against the sign of r, and state the final answer clearly.

Quickest way: Shortcut with deviations from the means

When to use it: Use when raw X and Y values are given with few items, or when the means are whole numbers.

Compute X̄ and Ȳ first.
Write the deviations x and y in two columns, then x², y² and xy.
Add the columns to get Σx², Σy² and Σxy. Check that Σx and Σy are zero.
Divide: b_yx = Σxy ÷ Σx², b_xy = Σxy ÷ Σy².
Write the line, substitute, and check that r = √(b_yx × b_xy) is not above 1.

Common mistakes in Simple Linear Regression Analysis

Using the wrong standard deviation ratio, such as b_yx = r σx ÷ σy.
The subscripts look alike and students memorise without logic.
Fix: Put the dependent variable's standard deviation on top. For Y on X, Y is dependent, so b_yx = r σy ÷ σx.
Using the line of Y on X to estimate X from a given Y.
Students rearrange one equation instead of using the other line.
Fix: Use the line of X on Y to estimate X. Rearranging Y on X gives a different, wrong answer unless r = ±1.
Taking r as positive when both coefficients are negative.
The square root gives a positive number and the sign is forgotten.
Fix: Give r the common sign of b_yx and b_xy.
Mixing up which of two given equations is Y on X.
Equations are given in unlabelled form.
Fix: Assume one choice, find both coefficients, and check b_yx × b_xy ≤ 1. If it exceeds 1, swap the assumption.
Taking means from the equations by guesswork.
Students forget that both lines pass through (X̄, Ȳ).
Fix: Solve the two equations simultaneously. The solution is X̄ and Ȳ.
Errors in the raw-data formula, using nΣX² − (ΣX)² for both coefficients.
The numerator is the same, so the denominator is overlooked.
Fix: For b_yx the denominator uses X. For b_xy it uses Y.

Worked examples

Example 1

For 5 months, machine hours (X) and overhead cost in ₹ thousand (Y) are: X = 2, 4, 6, 8, 10 and Y = 7, 9, 13, 14, 17. Find the regression line of Y on X and estimate overhead for 7 machine hours.

Show the solution

n = 5. ΣX = 30, so X̄ = 6. ΣY = 60, so Ȳ = 12.
Deviations x = −4, −2, 0, 2, 4. Deviations y = −5, −3, 1, 2, 5.
Σx² = 16 + 4 + 0 + 4 + 16 = 40.
Σxy = 20 + 6 + 0 + 4 + 20 = 50.
b_yx = 50 ÷ 40 = 1.25.
Line: Y − 12 = 1.25 (X − 6), so Y = 4.5 + 1.25X.
For X = 7: Y = 4.5 + 8.75 = 13.25.

Answer: Y = 4.5 + 1.25X. Estimated overhead for 7 machine hours is ₹13.25 thousand, that is ₹13,250.

Example 2

The two regression equations are 8X − 10Y + 66 = 0 and 40X − 18Y = 214. Find the means of X and Y and the correlation coefficient.

Show the solution

Both lines pass through the means. Solve them together.
From the first: 8X = 10Y − 66, so X = 1.25Y − 8.25.
Put in the second: 40(1.25Y − 8.25) − 18Y = 214.
50Y − 330 − 18Y = 214, so 32Y = 544, and Y = 17.
Then X = 1.25 × 17 − 8.25 = 21.25 − 8.25 = 13.
So X̄ = 13 and Ȳ = 17.
Assume the first is Y on X: Y = 6.6 + 0.8X... check: 10Y = 8X + 66, so b_yx = 0.8.
Second as X on Y: 40X = 18Y + 214, so b_xy = 18 ÷ 40 = 0.45.
Product = 0.8 × 0.45 = 0.36, which is ≤ 1, so the assumption is valid.
r = √0.36 = 0.6. Both coefficients are positive, so r is positive.

Answer: X̄ = 13, Ȳ = 17 and r = +0.6.

Exam tips

In MCQs, the properties are tested directly: same sign, product ≤ 1, r as the geometric mean. Learn them well.
When two equations are given, always check b_yx × b_xy ≤ 1 before finding r. This protects you from labelling errors.
Show the means, the coefficient and the final line separately. Marks are given for each step even if the arithmetic slips.
State the estimate with units and say that it is an average estimate, valid only within the range of observed data.
Do not mix up correlation and regression in theory answers. Give at least three clear differences.

Practice questions from Business Forecasting Models - Time Series and Regression Analysis

Simple Linear Regression Analysis in other exams

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

Simple Linear Regression Analysis: frequently asked questions

What is the difference between the regression line of Y on X and X on Y?

The line of Y on X estimates Y for a given X and minimises the squared vertical errors. The line of X on Y estimates X for a given Y and minimises the squared horizontal errors. They are different lines that cross at the means, and coincide only when r = ±1.

How do you calculate the regression coefficient?

With r and standard deviations, use b_yx = r σy ÷ σx and b_xy = r σx ÷ σy. With raw data, use b_yx = [nΣXY − ΣXΣY] ÷ [nΣX² − (ΣX)²]. For b_xy, replace the denominator with nΣY² − (ΣY)².

What are the properties of regression coefficients?

Both coefficients have the same sign, and that sign is the sign of r. Their product equals r², so it cannot exceed 1. The coefficients do not change with a change of origin but do change with a change of scale.

What is the difference between correlation and regression?

Correlation measures the strength and direction of a linear relationship and is symmetric between X and Y. Regression gives an equation to estimate one variable from the other and is not symmetric. Correlation is a pure number from −1 to +1, while regression coefficients have units.