Study guide
2.4 Scattergraph and Least-Squares Regression for Cost Estimation
CIMA BA2: Fundamentals of Management Accounting Exam
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1 Chapter 1: The Context of Management Accounting
1.1 The Need for Management Accounting, the GMAP Definition & Comparison with Financial Accounting 1.2 Qualities of Good Management Information: From Data to Decisions 1.3 Information for Operational, Managerial and Strategic Levels 1.4 The Role of the Management Accountant: CIMA's View, IFAC's Domain & Position in the Organisation 1.5 CIMA as a Professional Body: Members, Candidates, the Profession & Society
2 Chapter 2: Cost Identification, Classification and Behaviour
2.1 Cost Classification by Nature, Function, and Element 2.2 Cost Behaviour Patterns and Relevant Range 2.3 The High-Low Method and Cost Prediction 2.4 Scattergraph and Least-Squares Regression for Cost Estimation
3 Chapter 3: Overheads and Absorption Costing
3.1 Production Overheads: Allocation and Apportionment 3.2 Overhead Absorption Rates (OAR) & Under/Over-Absorption
4 Chapter 4: Marginal Costing, Profit Reconciliation and Cost-Based Pricing
4.1 Marginal Costing vs Absorption Costing Principles 4.2 Reconciling Marginal and Absorption Costing Operating Profits 4.3 Pricing with Cost Information: Full-Cost-Plus, Marginal-Cost-Plus, Mark-Ups, Margins and Return Targets
5 Chapter 5: Budget Preparation: Functional, Master and Cash Budgets
5.1 Purposes of Budgeting & The Budgeting Process 5.2 The Principal Budget Factor & Functional Budgets 5.3 The Master Budget: Budgeted Profit or Loss, Financial Position and Cash Flow 5.4 Cash Budgets & Cash Flow Forecasting
6 Chapter 6: Budgetary Control: Flexible Budgets, Behaviour and Budgeting Approaches
6.1 Fixed vs Flexible Budgeting and Budget Flexing 6.2 Budgetary Control Systems, Participation & Behavioural Aspects 6.3 Alternative Budgeting Approaches: Incremental, ZBB & Rolling Budgets
7 Chapter 7: Standard Costing and Cost Variances
7.1 Standard Costing Principles and Types of Standards 7.2 Direct Material Price and Usage Variances 7.3 Direct Labour Rate, Efficiency, and Idle Time Variances 7.4 Variable and Fixed Overhead Variances
8 Chapter 8: Sales Variances, Operating Statements and Interpreting Variances
8.1 Sales Variances and Operating Statements (Marginal and Absorption Formats) 8.2 Interpreting Variances: Causes, Interrelationships and Deciding What to Investigate
9 Chapter 9: Performance Measurement
9.1 The Need for Performance Measures: Responsibility Accounting, Controllability and Service Industries 9.2 Financial and Non-Financial Performance Measures in Manufacturing and Service Organisations
10 Chapter 10: Integrated Accounts, Job and Batch Costing and Management Reports
10.1 Integrating Cost and Financial Accounts: Control Accounts and the Manufacturing Account 10.2 Integrated Ledgers with Under- and Over-Absorption and Standard Cost Variances 10.3 Job and Batch Costing: Job Cost Cards, Job Accounts and Pricing a Job 10.4 Cost Reports for Manufacturing, Service and Not-for-Profit Organisations
11 Chapter 11: Risk, Probability and Statistics for Decision-Making
11.1 Risk, Uncertainty and Probability: Rules and Joint Probabilities for Decisions 11.2 Expected Values, Expected Value Tables and Their Limitations 11.3 Averages and Dispersion: Mean, Median, Mode, Range, Variance, Standard Deviation and Coefficient of Variation 11.4 The Normal Distribution: Z-Scores, Normal Distribution Tables and Business Probabilities
12 Chapter 12: Relevant Costing and Short-Term Decisions
12.1 Relevant Cost Concepts: Future, Incremental, and Opportunity Costs 12.2 Relevant Costs of Materials, Labour, and Overheads 12.3 Short-Term Operational Decisions: Make-or-Buy, Shutdowns, Special Orders
13 Chapter 13: Breakeven Analysis and Limiting Factor Analysis
13.1 Single-Product Break-Even Analysis & Margin of Safety 13.2 Target Profit Calculations, Operating Gearing & CVP Assumptions 13.3 Multi-Product CVP Analysis & Break-Even Charts 13.4 Limiting Factor Analysis (Key Factor Analysis)
14 Chapter 14: Long-Term Decisions: Time Value of Money, Payback, NPV and IRR
14.1 Time Value of Money: Compounding, Discounting, Annuities & Perpetuities 14.2 Payback Period Method: Constant and Uneven Cash Flows 14.3 Net Present Value (NPV): Methodology and Decision Rules 14.4 Internal Rate of Return (IRR) and Interpolation
2.4 Scattergraph and Least-Squares Regression for Cost Estimation
Key Takeaways Explain This Topic AI
A scattergraph plots every observed pair of activity and cost; the line of best fit's intercept estimates fixed cost and its gradient estimates variable cost per unit.
Least-squares regression calculates b = (nΣxy − ΣxΣy) ÷ (nΣx² − (Σx)²) and a = ȳ − b x̄, using all observations rather than only the highest and lowest.
In the cost equation y = a + bx, the intercept a is the estimated fixed cost and the slope b is the estimated variable cost per unit of activity.
Regression is objective and uses all data, but it still assumes a linear relationship and gives reliable predictions only within the range of activity observed.
Last updated: September 2026
Why this topic is examined
Syllabus area B1(c) asks you to calculate appropriate costs having identified cost behaviour , using "high-low, graphical and regression analysis methods to establish and predict total cost". Section 2.3 covered the high-low method. This section covers the other two. Expect number-entry questions that give you summary totals (Σx, Σy, Σxy, Σx²) and ask for the variable cost per unit, the fixed cost or a cost forecast, plus conceptual questions about which method is more reliable.
All three methods estimate the same cost equation: y = a + b x y = a + bx y = a + b x
where y y y is total cost, x x x is the activity level, a a a is the fixed cost (intercept) and b b b is the variable cost per unit of activity (slope).
- The scattergraph (graphical) method
- How it works
Plot each period's observation on a graph, with activity on the horizontal axis and total cost on the vertical axis .
Look at the pattern of points. If they lie roughly along a straight line, a linear cost model is reasonable.
Draw a line of best fit by eye , so that the points are spread evenly above and below it.
Read off the intercept where the line meets the cost axis: this is the estimated fixed cost .
Measure the gradient of the line (change in cost divided by change in activity): this is the estimated variable cost per unit .
Strengths and weaknesses
- Strengths Weaknesses
- Uses every observation, not just two The line is drawn by judgement, so two people may draw different lines
- Shows visually whether the relationship looks linear Reading the intercept and slope from a graph is imprecise
- Makes outliers (abnormal periods) easy to spot and exclude Hard to use when points are widely scattered
Note
The scattergraph's great practical value is diagnostic: it shows whether a straight-line model is sensible at all, and whether any period (for example, a month with a strike) should be excluded before using high-low or regression.
Least-squares linear regression
Regression analysis calculates the line of best fit mathematically. The least-squares method chooses the values of a a a and b b b that minimise the sum of the squared vertical distances between the observed costs and the line. Because the calculation is mechanical, everyone using the same data gets the same answer.
The formulas b = n ∑ x y − ∑ x ∑ y n ∑ x 2 − ( ∑ x ) 2 b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - \left(\sum x\right)^2} b = n ∑ x 2 − ( ∑ x ) 2 n ∑ x y − ∑ x ∑ y a = y ˉ − b x ˉ = ∑ y n − b ∑ x n a = \bar{y} - b\bar{x} = \frac{\sum y}{n} - b\frac{\sum x}{n} a = y ˉ − b x ˉ = n ∑ y − b n ∑ x
where n n n is the number of pairs of observations. CIMA provides relevant formulae and tables in the assessment, but you must know how to use them quickly and in the right order: calculate b b b first , then use it to find a a a .
Worked example
A maintenance department records five months of activity (thousands of machine hours) and cost ($000):
- Month Machine hours, 000 ( x x x ) Cost, $000 ( y y y ) x y xy x y x 2 x^2 x 2
- 1 1 6 6 1
- 2 2 8 16 4
- 3 3 11 33 9
- 4 4 12 48 16
- 5 5 13 65 25
- Total 15 50 168 55
Step 1: calculate the slope b b b . b = ( 5 × 168 ) − ( 15 × 50 ) ( 5 × 55 ) − 15 2 = 840 − 750 275 − 225 = 90 50 = 1.8 b = \frac{(5 \times 168) - (15 \times 50)}{(5 \times 55) - 15^2} = \frac{840 - 750}{275 - 225} = \frac{90}{50} = 1.8 b = ( 5 × 55 ) − 1 5 2 ( 5 × 168 ) − ( 15 × 50 ) = 275 − 225 840 − 750 = 50 90 = 1.8
Because y y y is in $000 and x x x is in thousands of hours, b = 1.8 b = 1.8 b = 1.8 means $1.80 of variable cost per machine hour .
Step 2: calculate the intercept a a a . a = 50 5 − ( 1.8 × 15 5 ) = 10 − 5.4 = 4.6 a = \frac{50}{5} - \left(1.8 \times \frac{15}{5}\right) = 10 - 5.4 = 4.6 a = 5 50 − ( 1.8 × 5 15 ) = 10 − 5.4 = 4.6
So the estimated fixed cost is $4,600 per month .
Step 3: write the cost equation and forecast. Total cost ($000) = 4.6 + 1.8 x \text{Total cost (\textdollar 000)} = 4.6 + 1.8x Total cost ($000) = 4.6 + 1.8 x
For a month with 3,500 machine hours ( x = 3.5 x = 3.5 x = 3.5 ): 4.6 + ( 1.8 × 3.5 ) = 4.6 + 6.3 = 10.9 ($000) 4.6 + (1.8 \times 3.5) = 4.6 + 6.3 = 10.9 \text{ (\textdollar 000)} 4.6 + ( 1.8 × 3.5 ) = 4.6 + 6.3 = 10.9 ($000)
The forecast cost is $10,900 .
Comparison with high-low on the same data
The high-low method would use only months 5 and 1: b = 13 − 6 5 − 1 = 1.75 a = 6 − ( 1.75 × 1 ) = 4.25 b = \frac{13 - 6}{5 - 1} = 1.75 \qquad a = 6 - (1.75 \times 1) = 4.25 b = 5 − 1 13 − 6 = 1.75 a = 6 − ( 1.75 × 1 ) = 4.25
High-low gives $1.75 per hour and $4,250 fixed, noticeably different from regression, because it ignores months 2, 3 and 4. Regression is generally the more reliable estimate when you have several observations.
How good is the fit?
The correlation coefficient r r r (covered in more depth in BA1) measures how closely the points follow a straight line, from − 1 -1 − 1 to + 1 +1 + 1 . Its square, the coefficient of determination r 2 r^2 r 2 , gives the proportion of the variation in cost explained by the variation in activity. For the example above: r = n ∑ x y − ∑ x ∑ y ( n ∑ x 2 − ( ∑ x ) 2 ) ( n ∑ y 2 − ( ∑ y ) 2 ) = 90 50 × 170 ≈ 0.976 r = \frac{n\sum xy - \sum x\sum y}{\sqrt{\left(n\sum x^2 - (\sum x)^2\right)\left(n\sum y^2 - (\sum y)^2\right)}} = \frac{90}{\sqrt{50 \times 170}} \approx 0.976 r = ( n ∑ x 2 − ( ∑ x ) 2 ) ( n ∑ y 2 − ( ∑ y ) 2 ) n ∑ x y − ∑ x ∑ y = 50 × 170 90 ≈ 0.976
(using ∑ y 2 = 534 \sum y^2 = 534 ∑ y 2 = 534 ). So r 2 ≈ 0.953 r^2 \approx 0.953 r 2 ≈ 0.953 : about 95% of the month-to-month variation in maintenance cost is explained by machine hours. A high r 2 r^2 r 2 supports using the equation for budgeting; a low one suggests other cost drivers matter.
Choosing between the three methods
- Feature High-low Scattergraph Regression
- Observations used Two All All
- Objectivity Objective Subjective (drawn by eye) Objective
- Effort Very quick Quick More calculation
Sensitivity to one abnormal period High, if it is the high or low point Low, if the outlier is spotted Moderate
- Shows whether linear model is suitable No Yes Partly, via r r r and r 2 r^2 r 2
- Limitations common to all three
They assume cost behaviour is linear within the range observed.
Forecasts are reliable only within the relevant range . Using the equation for 20,000 hours when the data covered 1,000 to 5,000 hours is extrapolation, and the relationship may not hold.
They rely on past data , which may not reflect future prices, technology or efficiency. Adjust for inflation (as shown in Section 2.3) before analysing costs from different years.
Correlation does not prove causation. A strong r r r shows association; the management accountant should confirm that the activity measure genuinely drives the cost.