Performance Management · Analytical techniques in budgeting and forecasting

Linear Regression Analysis for ACCA Performance Management

Updated 11 October 2026 · Fact-checked

Linear regression fits the straight line y = a + bx that best matches past data, using least squares. The value a estimates fixed cost and b estimates variable cost per unit. Work out b first, then a, then substitute a value of x to forecast y. Check reliability with r².

Understand Linear Regression Analysis

Cost data rarely lies on a perfect straight line. Costs move up and down for reasons other than activity. Linear regression finds the single straight line that fits all the points as closely as possible.

The line is y = a + bx. Here y is the dependent variable (usually total cost). x is the independent variable (usually activity, such as units or machine hours). a is the intercept, which estimates fixed cost. b is the slope, which estimates variable cost per unit of x.

The method is called least squares. For each point, measure the vertical gap between the actual y and the line. Square each gap and add them up. The best line is the one that makes this total as small as possible. You do not need to prove this. You only need to apply the formulas.

The high-low method uses only two points, the highest and lowest activity levels. Regression uses every data point, so it is usually more reliable. It also gives you r², which tells you how well the line fits.

Regression is a forecasting tool, not a guarantee. It assumes a linear relationship and that past behaviour continues. Forecasts inside the range of your data (interpolation) are more reliable than forecasts outside it (extrapolation).

Key rules to remember

Regression line: y = a + bx; a = fixed cost, b = variable cost per unit of activity, x = activity level.
Slope b: b = (nΣxy − ΣxΣy) ÷ (nΣx² − (Σx)²); n is the number of pairs of data. Calculate b before a. Note that (Σx)² is the square of the total, not Σx².
Intercept a: a = (Σy − bΣx) ÷ n; Equivalent to a = ȳ − b x̄, where ȳ and x̄ are the means.
Correlation coefficient: r = (nΣxy − ΣxΣy) ÷ √[(nΣx² − (Σx)²)(nΣy² − (Σy)²)]; Ranges from −1 to +1. Close to +1 or −1 means a strong linear relationship.
Coefficient of determination: r² = proportion of variation in y explained by variation in x; If r = 0.9, then r² = 0.81, so 81% of the variation in y is explained by x. The rest is due to other factors.

How to solve Linear Regression Analysis questions

Use this order for any regression question. Check first whether the question already gives you the sums or the equation.

1Identify y (the cost or value to forecast) and x (the activity or driver). Note the units of each, for example thousands.
2If Σx, Σy, Σxy and Σx² are not given, build a table with columns x, y, xy and x², and total each column. Count n.
3Calculate b using b = (nΣxy − ΣxΣy) ÷ (nΣx² − (Σx)²). Do the top and bottom lines separately.
4Calculate a using a = (Σy − bΣx) ÷ n.
5Write the equation y = a + bx. State what a and b mean in context, with units: fixed cost and variable cost per unit.
6Substitute the required x to forecast y. Convert back from thousands if needed.
7Comment on reliability: size of r or r², whether x is inside the data range, and whether the linear assumption is reasonable.

Quickest way: Calculator route using the sums

When to use it: Use when the exam gives Σx, Σy, Σxy and Σx², or when you have a small data set and need speed in the computer-based exam.

Write n, Σx, Σy, Σxy and Σx² on your workings sheet.
Compute the top line nΣxy − ΣxΣy, then the bottom line nΣx² − (Σx)².
Divide to get b. Keep the full figure in your calculator memory.
Compute a from Σy − bΣx, divided by n.
Sanity check: the line must pass through the means, so a + b x̄ should equal ȳ.
For objective questions, check which option matches your a and b before you spend time on anything else.

Common mistakes in Linear Regression Analysis

Confusing Σx² with (Σx)².
The notation looks similar and both appear in the denominator.
Fix: Σx² is the total of each x squared. (Σx)² is the total of x, then squared. Calculate them separately and label them.
Calculating a before b.
The equation is written as a + bx, so students start with a.
Fix: The formula for a needs b. Always find b first.
Ignoring units such as thousands.
Students work with the table figures and forget the scale.
Fix: Write the units next to a and b. If x is in 000 units and y in $000, then b is $ per unit and a is in $000. Convert before giving the final answer.
Swapping x and y.
The question does not say which is which.
Fix: Cost depends on activity, so cost is y and activity is x. Activity is the driver.
Forecasting far outside the data range and presenting it as reliable.
The equation will give a number for any x.
Fix: Give the figure, then state that extrapolation is less reliable because the relationship may not hold outside the observed range.
Treating r² as r, or saying a high r² proves cause.
Both are called measures of correlation.
Fix: r² is the proportion of variation in y explained by x. A high value shows a strong linear fit, not that x causes y.

Worked examples

Example 1

A company records output (x, in thousands of units) and total cost (y, in $000) for five months: (1, 14), (2, 18), (3, 21), (4, 26), (5, 29). Calculate the regression line and forecast the total cost at 4,500 units.

Show the solution

n = 5. Σx = 1+2+3+4+5 = 15. Σy = 14+18+21+26+29 = 108.
Σxy = 14 + 36 + 63 + 104 + 145 = 362. Σx² = 1+4+9+16+25 = 55.
b = (5 × 362 − 15 × 108) ÷ (5 × 55 − 15²) = (1,810 − 1,620) ÷ (275 − 225) = 190 ÷ 50 = 3.8.
a = (108 − 3.8 × 15) ÷ 5 = (108 − 57) ÷ 5 = 10.2.
Equation: y = 10.2 + 3.8x. Fixed cost is $10,200 and variable cost is $3.80 per unit.
For 4,500 units, x = 4.5. y = 10.2 + 3.8 × 4.5 = 10.2 + 17.1 = 27.3, which is $27,300.
4.5 lies inside the data range of 1 to 5, so this is interpolation and is reasonably reliable.

Answer: y = 10.2 + 3.8x. Fixed cost is $10,200, variable cost is $3.80 per unit, and the forecast total cost at 4,500 units is $27,300.

Example 2

Six months of data on machine hours (x, in 000 hours) and overhead cost (y, in $000) give: Σx = 90, Σy = 1,200, Σxy = 19,300, Σx² = 1,450. The correlation coefficient r is 0.9. (a) Find the regression equation. (b) Forecast overhead at 18,000 machine hours. (c) Comment on reliability.

Show the solution

n = 6.
b = (6 × 19,300 − 90 × 1,200) ÷ (6 × 1,450 − 90²) = (115,800 − 108,000) ÷ (8,700 − 8,100) = 7,800 ÷ 600 = 13.
a = (1,200 − 13 × 90) ÷ 6 = (1,200 − 1,170) ÷ 6 = 5.
Equation: y = 5 + 13x. Fixed overhead is $5,000 and variable overhead is $13 per machine hour.
For 18,000 hours, x = 18. y = 5 + 13 × 18 = 5 + 234 = 239, which is $239,000.
r² = 0.9² = 0.81, so 81% of the variation in overhead is explained by machine hours. The fit is strong.
Check: the mean x is 15 and the mean y is 200. 5 + 13 × 15 = 200, which confirms the line.

Answer: (a) y = 5 + 13x. (b) $239,000. (c) r² = 0.81 shows a strong linear fit, but the forecast is reliable only if 18,000 hours is within the observed range and past cost behaviour continues.

Exam tips

In Section A and B objective questions, the sums are usually given. Go straight to the formulas for b and then a. Wrong answers score zero, so check arithmetic once.
In Section C, show the formula, the substituted numbers and the result. Method marks are available even if you make an arithmetic slip.
Always interpret a and b in words with units. Examiners often reward the statement that a is fixed cost and b is variable cost per unit.
Be ready to compare with the high-low method: regression uses all data points and gives r², while high-low uses only two points and can be distorted by outliers.
When asked to comment on a forecast, mention r or r², interpolation versus extrapolation, and the assumption that the relationship is linear and stable.

Practice questions from Analytical techniques in budgeting and forecasting

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.

Linear Regression Analysis: frequently asked questions

How do I calculate a and b in least squares regression?

Find b first using b = (nΣxy − ΣxΣy) ÷ (nΣx² − (Σx)²). Then find a using a = (Σy − bΣx) ÷ n. Substitute both into y = a + bx.

What is the difference between the high-low method and regression analysis?

The high-low method uses only the highest and lowest activity points, so one unusual point can distort the result. Regression uses all the data and gives a line of best fit. It also lets you measure reliability with r and r².

What do a and b mean in y = a + bx?

a is the intercept and estimates fixed cost, the cost when activity is zero. b is the slope and estimates the variable cost for each extra unit of activity. Always state units.

Do I need to memorise the regression formulas for ACCA PM?

You should learn them and practise them, because you need them to solve the questions. Check the formulae sheet provided in your exam, but do not rely on it without practice. Know how to apply the formulas quickly.