2.4 Scattergraph and Least-Squares Regression for Cost Estimation

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:

$$y = a + bx$$

where $y$ is total cost, $x$ is the activity level, $a$ is the fixed cost (intercept) and $b$ is the variable cost per unit of activity (slope).

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The scattergraph (graphical) method

How it works

1. Plot each period's observation on a graph, with activity on the horizontal axis and total cost on the vertical axis.
2. Look at the pattern of points. If they lie roughly along a straight line, a linear cost model is reasonable.
3. Draw a line of best fit by eye, so that the points are spread evenly above and below it.
4. Read off the intercept where the line meets the cost axis: this is the estimated fixed cost.
5. 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

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Least-squares linear regression

Regression analysis calculates the line of best fit mathematically. The least-squares method chooses the values of $a$ and $b$ 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

$$b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - \left(\sum x\right)^2}$$ $$a = \bar{y} - b\bar{x} = \frac{\sum y}{n} - b\frac{\sum x}{n}$$

where $n$ 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 $b$ first, then use it to find $a$.

Worked example

A maintenance department records five months of activity (thousands of machine hours) and cost ($000):

Step 1: calculate the slope $b$.

$$b = \frac{(5 \times 168) - (15 \times 50)}{(5 \times 55) - 15^2} = \frac{840 - 750}{275 - 225} = \frac{90}{50} = 1.8$$

Because $y$ is in $000 and $x$ is in thousands of hours, $b = 1.8$ means $1.80 of variable cost per machine hour.

Step 2: calculate the intercept $a$.

$$a = \frac{50}{5} - \left(1.8 \times \frac{15}{5}\right) = 10 - 5.4 = 4.6$$

So the estimated fixed cost is $4,600 per month.

Step 3: write the cost equation and forecast.

$$\text{Total cost (\textdollar 000)} = 4.6 + 1.8x$$

For a month with 3,500 machine hours ($x = 3.5$):

$$4.6 + (1.8 \times 3.5) = 4.6 + 6.3 = 10.9 \text{ (\textdollar 000)}$$

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:

$$b = \frac{13 - 6}{5 - 1} = 1.75 \qquad a = 6 - (1.75 \times 1) = 4.25$$

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.

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How good is the fit?

The correlation coefficient $r$ (covered in more depth in BA1) measures how closely the points follow a straight line, from $-1$ to $+1$. Its square, the coefficient of determination $r^2$, gives the proportion of the variation in cost explained by the variation in activity. For the example above:

$$r = \frac{n\sum xy - \sum x\sum y}{\sqrt{\left(n\sum x^2 - (\sum x)^2\right)\left(n\sum y^2 - (\sum y)^2\right)}} = \frac{90}{\sqrt{50 \times 170}} \approx 0.976$$

(using $\sum y^2 = 534$). So $r^2 \approx 0.953$: about 95% of the month-to-month variation in maintenance cost is explained by machine hours. A high $r^2$ supports using the equation for budgeting; a low one suggests other cost drivers matter.

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Choosing between the three methods

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 $r$ shows association; the management accountant should confirm that the activity measure genuinely drives the cost.