Performance Management · Analytical techniques in budgeting and forecasting

Correlation Coefficient and Coefficient of Determination in ACCA PM

Updated 11 October 2026 · Fact-checked

The correlation coefficient (r) measures the strength and direction of a linear relationship between two variables, from -1 to +1. The coefficient of determination (r²) is r squared. It shows the proportion of variation in y explained by x. A higher r² means forecasts from the regression line are more reliable.

Understand Correlation Coefficient and Coefficient of Determination

When you forecast costs or sales, you often assume one variable (y) depends on another (x). For example, overheads may depend on machine hours. Regression gives you a line. Correlation tells you how well that line fits the data.

The correlation coefficient, r, always lies between -1 and +1. A value near +1 means a strong positive linear relationship: as x rises, y rises. A value near -1 means a strong negative linear relationship: as x rises, y falls. A value near 0 means little or no linear relationship. The sign of r is the same as the sign of the slope b in the regression line.

The coefficient of determination, r², is r multiplied by itself. It is always between 0 and 1 and is usually quoted as a percentage. If r² = 0.81, then 81% of the variation in y is explained by variation in x. The other 19% is due to other factors or random variation.

The higher r², the more reliable the forecast. But r only measures a linear relationship. A curved relationship can give a low r even when the link is strong. Also, correlation is not causation. Two variables can move together without one causing the other.

Reliability also depends on the data. A small sample, a forecast outside the range of the data (extrapolation), or a relationship that changes over time all weaken a forecast, even if r is high.

Key rules to remember

Correlation coefficient: r = (nΣxy − ΣxΣy) ÷ √[(nΣx² − (Σx)²)(nΣy² − (Σy)²)]; This is given in the ACCA formulae sheet, but you must know how to use it. n is the number of pairs of data.
Coefficient of determination: r² = r × r; Gives the proportion of variation in y explained by x. Express it as a percentage when you comment.
Range of r: -1 ≤ r ≤ +1; +1 is perfect positive correlation, -1 is perfect negative, 0 means no linear correlation.
Unexplained variation: 1 − r²; The proportion of variation in y caused by other factors or chance.
Regression line (for linking): y = a + bx; The sign of b matches the sign of r.

How to solve Correlation Coefficient and Coefficient of Determination questions

Use this method for calculation and interpretation questions on r and r².

1Identify x (the independent variable) and y (the dependent variable). Count the pairs of data, n.
2Build columns for x, y, xy, x² and y². Total each column to get Σx, Σy, Σxy, Σx² and Σy².
3Work out the numerator: nΣxy − ΣxΣy. Its sign tells you the sign of r.
4Work out the two parts of the denominator: nΣx² − (Σx)² and nΣy² − (Σy)². Multiply them and take the square root.
5Divide the numerator by the denominator to get r. Check it lies between -1 and +1.
6Square r to get r². Convert it to a percentage.
7Interpret in words: strength, direction, and the share of variation explained. State what is left unexplained.
8Comment on reliability: sample size, extrapolation outside the data range, and whether the relationship is truly linear.

Quickest way: Shortcut for exam time pressure

When to use it: Use this in Section A or B objective questions, where r or r² is given and you only need to interpret it or judge a forecast.

Read the sign of r. Positive means y rises with x. Negative means y falls as x rises.
Square r mentally to get r². Read it as the percentage of variation in y explained by x.
Subtract r² from 1 for the unexplained part.
Check whether the forecast x value lies inside the data range. If not, the forecast is less reliable whatever r² is.
If you must calculate r, keep the totals on your scratch pad and compute the numerator and each denominator part separately before the square root.

Common mistakes in Correlation Coefficient and Coefficient of Determination

Saying r² = 0.64 means 64% correlation.
Students mix up r and r².
Fix: Say that 64% of the variation in y is explained by x. Then r = 0.8 (or -0.8).
Losing the negative sign when taking the square root of r².
A square root gives only a positive value on a calculator.
Fix: Take the sign from the slope b or from the numerator nΣxy − ΣxΣy.
Using (Σx)² and Σx² as if they were the same.
The notation looks similar and is rushed.
Fix: Σx² means square each x, then add. (Σx)² means add first, then square.
Claiming that high correlation proves x causes y.
A strong number feels like proof.
Fix: Say that the variables move together, but another factor or chance may explain it.
Saying a forecast is reliable just because r² is high.
Students ignore where the forecast is made.
Fix: Also check for extrapolation, small sample size, and whether the past relationship will continue.
Treating r near 0 as meaning no relationship at all.
Students forget that r measures only linear relationships.
Fix: Say there is little linear correlation. A non-linear relationship could still exist.

Worked examples

Example 1

A company records advertising spend (x, in $000) and sales (y, in $00,000) over five months. Σx = 15, Σy = 27, Σxy = 93, Σx² = 55, Σy² = 163, n = 5. Calculate r and r², and interpret them.

Show the solution

Numerator = nΣxy − ΣxΣy = (5 × 93) − (15 × 27) = 465 − 405 = 60.
First denominator part = nΣx² − (Σx)² = (5 × 55) − 225 = 275 − 225 = 50.
Second denominator part = nΣy² − (Σy)² = (5 × 163) − 729 = 815 − 729 = 86.
Denominator = √(50 × 86) = √4,300 = 65.57.
r = 60 ÷ 65.57 = 0.915.
r² = 0.915 × 0.915 = 0.837 (or 3,600 ÷ 4,300 = 0.837).
Interpretation: there is a strong positive linear relationship. About 84% of the variation in sales is explained by advertising spend. About 16% is due to other factors.

Answer: r = 0.915 and r² = 0.837. Strong positive correlation; about 84% of the variation in sales is explained by advertising spend.

Example 2

A regression of overhead cost (y, in $) on machine hours (x) gives y = 12,000 + 4.5x, with r = 0.85. The data used machine hours between 1,000 and 3,000. The manager wants forecasts for 2,500 hours and 5,000 hours. Calculate both forecasts and comment on their reliability.

Show the solution

Forecast for 2,500 hours: y = 12,000 + (4.5 × 2,500) = 12,000 + 11,250 = $23,250.
Forecast for 5,000 hours: y = 12,000 + (4.5 × 5,000) = 12,000 + 22,500 = $34,500.
Work out r² = 0.85 × 0.85 = 0.7225, or about 72%. So 72% of the variation in overheads is explained by machine hours. About 28% is due to other factors.
2,500 hours is within the data range 1,000 to 3,000. This is interpolation, so the forecast is reasonably reliable. The strong r supports this, though 28% is unexplained.
5,000 hours is outside the data range. This is extrapolation. The cost behaviour may change at higher activity, for example step costs or capacity limits. The forecast is much less reliable despite the same r².

Answer: Forecasts are $23,250 (2,500 hours) and $34,500 (5,000 hours). r² is about 72%. The first forecast is reasonably reliable; the second is extrapolated and less reliable.

Exam tips

In objective questions, a given r² is often used to test the sentence you pick. Choose the option that says 'proportion of variation in y explained by x', not 'strength of correlation'.
Write the numerator, each denominator part and the square root as separate lines in Section C. You can earn method marks even if you make an arithmetic slip.
When asked to comment on reliability, give at least three points: size of r², sample size, and extrapolation. Add that correlation is not causation if the question mentions cause.
Always state the direction (positive or negative) as well as the strength when you interpret r.
Check that your r is between -1 and +1. A value outside this range means an arithmetic error, usually in the denominator.

Practice questions from Analytical techniques in budgeting and forecasting

Correlation Coefficient and Coefficient of Determination in other exams

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

Correlation Coefficient and Coefficient of Determination: frequently asked questions

What is the difference between r and r²?

r measures the strength and direction of a linear relationship, from -1 to +1. r² is r squared and shows the proportion of variation in y explained by x. r² is never negative.

What is a good value of r or r² in ACCA PM?

ACCA does not set a fixed cut-off. A value of r close to +1 or -1 means a strong linear relationship, and a high r² means forecasts are more reliable. In your answer, say how strong it is and then qualify it, for example with sample size or extrapolation.

Do I need to memorise the correlation formula?

The formula appears in the ACCA formulae sheet. You still need to know how to work out each total and apply it quickly. You must also be able to interpret the result.

Can r² be used to prove that x causes y?

No. r² shows how much of the variation moves with x, not why. A third factor or chance could be the cause, so say that the relationship is statistical, not proof of cause.