Semi-Variable Costs
Chapter 3 is about splitting a semi-variable cost into its fixed and variable parts, so that total cost can be predicted at any level of activity – syllabus outcome B1(c). It covers the three methods the syllabus names: the high-low method, the graphical (scattergraph) method and least squares regression.
It then explains how to measure how good a straight-line fit is, using Pearson’s correlation coefficient (r) and the coefficient of determination (r²). The formulae for regression and correlation are given in the examination; you must know what each symbol means and be able to use them.
1 Introduction
The chapter relates to semi-variable costs i.e. part fixed and part variable. It may be necessary for you in the examination to identify the fixed and variable elements and in this chapter we will revise the ‘high-low’ method and also explain Regression Analysis.
In every method the cost is written as a straight line, y = a + bx, where y is the total cost, a is the fixed cost per period, b is the variable cost per unit and x is the level of activity (for example, units produced).
2 High-Low Method
This is a quick and easy approach that estimates fixed and variable costs by comparing the highest and lowest activity levels.
Take the highest and the lowest levels of activity – the independent variable, x – not the highest and lowest costs. Cost depends on activity, so activity is the independent variable and cost (y) is the dependent variable. The two are not always the same months.
Electricity costs for the first 6 months of the year are as follows:
Units produced | Cost ($) | |
January | 340 | 2,260 |
February | 300 | 2,160 |
March | 380 | 2,320 |
April | 420 | 2,400 |
May | 400 | 2,300 |
June | 360 | 2,266 |
Calculate the fixed and variable costs using the high-low method.
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A step in the fixed costs
Sometimes you are told that fixed costs step up between the low and the high levels of activity. The step is not variable cost, so remove it before comparing the two levels.
Fixed costs are $8,000 a month below 4,000 units, but rise by $2,000 (to $10,000) at 4,000 units and above. Total costs were $26,000 at 3,000 units and $52,000 at 7,000 units.
Remove the step from the high level: $52,000 − $2,000 = $50,000. Then compare like with like: variable cost = ($50,000 − $26,000) ÷ (7,000 − 3,000) = $6 per unit.
Check the fixed cost at the low level: $26,000 − 3,000 × $6 = $8,000 ✓. The total cost of 5,000 units would be $10,000 + 5,000 × $6 = $40,000.
3 Problems with the high-low approach
It uses only two observations – the highest and the lowest – and ignores all the others.
If either of those two is unusual (a month when something went wrong, for example), the line, and so the fixed and variable costs, will be badly distorted. You cannot tell this just by looking at the two figures.
It assumes that the relationship between cost and activity is a straight line, and that fixed costs and the variable cost per unit stay the same across the whole range.
Its advantage is that it is quick and easy to calculate and to understand.
4 The graphical (scattergraph) method
The graphical method uses all the observations. Each one is plotted on a graph, with activity (x) on the horizontal axis and total cost (y) on the vertical axis. This is called a scattergraph. A line of best fit is then drawn by eye through the middle of the points, so that roughly as many points lie above the line as below it.
The fixed cost is read from the graph where the line crosses the vertical (cost) axis – the cost at nil activity.
The variable cost per unit is the slope of the line: take any two points on the line, and divide the difference in cost by the difference in activity.
The six months of electricity costs in Example 1 are plotted below, with a line of best fit drawn through them by eye and extended back to the cost axis.
The line crosses the cost axis at about $1,650, so the fixed cost is about $1,650 a month. At 400 units the line reads about $2,350, so the variable cost is about ($2,350 − $1,650) ÷ 400 = $1.75 per unit.
This is different from the high-low result ($1,560 fixed, $2 per unit) because the line takes account of all six months, not just the two extremes.
The graphical method is better than high-low because it uses every observation, and the graph shows at a glance whether the relationship looks like a straight line and whether any point is unusual. Its weakness is that the line is drawn by eye, so different people will draw slightly different lines and get slightly different answers. Regression analysis removes that judgement.
5 Regression
If there is a reasonable degree of linear correlation between two variables, we can use regression analysis to calculate the equation of the best fit for the data.
This is known as least squares linear regression.
It finds the straight line that most nearly passes through all the points: the line for which the total of the squared vertical distances between the points and the line is as small as possible.
If the equation relating two variables, x and y, is
y = a + bx
then the values of a and b may be calculated using the following formulae (which are given in the examination):
Formula | |
Variable cost per unit (slope) | b = (nΣxy − ΣxΣy) ÷ (nΣx² − (Σx)²) |
Fixed cost (intercept) | a = Σy ÷ n − b × Σx ÷ n |
You are not told what the symbols mean, so learn them:
Symbol | Meaning |
x | the independent variable (activity, e.g. units produced) |
y | the dependent variable (total cost) |
n | the number of pairs of observations (e.g. the number of months) |
Σ | “the sum of” – add up whatever follows it |
Σxy | multiply each x by its y, then add up the results |
Σx² | square each x, then add up the results |
(Σx)² | add up the x values, then square the total |
Work out b first, because the formula for a uses b. To keep the numbers small you may write the data in hundreds of units or thousands of dollars – but remember to convert back at the end.
The following table shows the number of units produced each month and the total cost incurred:
Units | Cost | |
January | 100 | 40 |
February | 400 | 65 |
March | 200 | 45 |
April | 700 | 80 |
May | 600 | 70 |
June | 500 | 70 |
July | 300 | 50 |
Calculate the regression line, y = a + bx
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6 Problems with regression analysis
The points are rarely exactly on a straight line (in Example 2, May and June have different output but the same cost), so the answer is only an estimate.
It assumes the relationship is linear. If the points really follow a curve, the formulae will still produce a straight line, but it will be meaningless. The correlation coefficient (section 7) tests how close to linear the relationship is.
It is based on past data, which may not be a reliable guide to the future if conditions (prices, methods, efficiency) change.
7 The correlation coefficient
Pearson’s correlation coefficient is a measure of how linear the relationship between variables is.
A correlation coefficient of +1 indicates perfect positive linear correlation, whereas -1 indicates perfect negative linear correlation.
The further away from + or – 1, the less linear correlation exists.
The correlation coefficient may be calculated using the following formula (which is given to you in the examination):
r = (nΣxy − ΣxΣy) ÷ √[(nΣx² − (Σx)²) × (nΣy² − (Σy)²)]
The only new term is Σy² – square each y and add up the results. The sign of r comes from the top line of the formula: if nΣxy − ΣxΣy is negative, r is negative.
Value of r | Meaning |
+1 | Perfect positive linear correlation – all points lie exactly on a straight line sloping upwards (as x rises, y rises) |
Close to +1 (e.g. +0.9) | Strong positive correlation – the points lie close to an upward-sloping line |
0 | No linear correlation – no straight-line link between the two |
Close to −1 (e.g. −0.9) | Strong negative correlation – as x rises, y falls, close to a straight line (for example, selling price and the number of units sold) |
−1 | Perfect negative linear correlation – all points lie exactly on a downward-sloping line |
r is always between −1 and +1. An answer outside that range means there is an arithmetic mistake.
Using the data in Example 2:
calculate the correlation coefficient, and explain what it shows
calculate the coefficient of determination (section 8) and explain what it means.
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8 Coefficient of determination
The coefficient of determination is the square of the coefficient of correlation (r2).
It is a measure of how much of the variation in the dependent variable is ‘explained’ by the variation of the independent variable.
No formula is given for it – simply square r. Because it is a square, it is always positive, and it is usually expressed as a percentage: r = 0.9 gives r² = 0.81, meaning 81% of the change in y is explained by the change in x.
9 Test your knowledge
Two quick checks before you move on: work through the flashcards to fix this chapter’s terms and definitions, then sit the objective questions for exam-style practice. Both mark themselves and explain the answers as you go.
Semi-Variable Costs
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