Splitting Mixed Costs: High–Low and Simple Cost Models

Learning outcomes

After studying this chapter, you will be able to:

• Separate a mixed (semi-variable) cost into estimated fixed and variable elements using the high–low method.
• Express cost behaviour as a simple linear relationshipy = a + bXand use it to forecast total cost at different activity levels.
• Evaluate whether the estimate is credible by considering the relevant range, outliers, step changes, and data quality.
• Decide when high–low is acceptable for planning and when a method using more of the data is more appropriate.

Overview & key concepts

Many overheads do not behave as purely fixed or purely variable. They include a standing charge plus an element that changes with activity. Examples include utilities, equipment support contracts, vehicle running costs, and some labour arrangements.

Splitting mixed costs is useful for:

• preparing budgets and forecasts that flex with activity
• estimating how total costs will change as volume changes
• supporting short-term decisions where cost behaviour matters

These techniques are for planning and analysis. They do not create accounting entries by themselves.

Core theory and frameworks

Notation used in this chapter

To keep your workings consistent:

• X= activity level (the chosen cost driver, measured in driver units)
• y= total mixed cost for the period
• a= fixed cost per period (the base cost)
• b= variable cost per unit of X

1) Identifying mixed costs

A mixed cost contains both:

• afixed component (a)that stays constant in total within a normal operating band, and
• avariable component (bX)that rises as activity rises.

A typical pattern is that total cost increases with activity, but it does not start from zero, because a fixed element is present.

Exam tip — common misconception: expenses and liabilities

Mixed costs are expenses for profit measurement. They only become a liability if unpaid at a reporting date (for example, accrued expenses or trade payables). The cost-splitting process itself does not automatically create liabilities.

2) Choosing a cost driver (X)

A cost driver is a measurable factor used to explain and predict changes in cost. Examples include units produced, machine-hours, labour-hours, miles driven, or number of deliveries.

The driver should have a sensible link to the cost. If the driver is poorly chosen, the model can produce neat numbers that are not useful.

3) High–low method (two-point estimate)

The high–low method uses the periods with the highest activity and lowest activity to estimate b and a.

Step 1: Variable cost per unit (b)

Compute the variable rate as:

b = change in total cost ÷ change in activity

b = (y_high - y_low) / (X_high - X_low)

Alternative delta notation:

b = Δy / ΔX

Step 2: Fixed cost per period (a)

Start from the linear model:

y = a + bX

Substitute either the high or low observation:

a = y_high - bX_high

or

a = y_low - bX_low

If the arithmetic is correct, both approaches should give the same a.

4) Constructing and using the cost equation

Once a and b are estimated:

y = a + bX

You can use this to forecast total cost at different activity levels, provided the forecast activity remains within a sensible operating range.

5) Relevant range, outliers, and step costs

Relevant range

The relevant range is the band of activity where it is reasonable to assume that:

• fixed cost (a) stays fixed in total, and
• the variable rate (b) is broadly stable.

Forecasts outside this band may be unreliable because capacity limits, overtime, or additional resources can change cost behaviour.

Outliers

An outlier is an observation that does not fit the general pattern (for example, one-off repairs, unusual waste, downtime, or temporary pricing changes). If either the high or low activity month is unusual, high–low can be distorted.

Step costs

A step cost remains fixed over a range of activity and then jumps to a new level once capacity is exceeded (for example, an additional supervisor or an extra equipment lease). A single straight-line model cannot capture step changes well.

6) Alternatives to high–low

High–low is best treated as a quick estimate. When more data is available, stronger approaches include:

• Scattergraph / visual fit(plot X against y and judge whether a straight line is reasonable)
• Least squares regression(uses all observations to estimate a and b, usually giving a more reliable fit than two-point methods)

Worked example

Narrative scenario

A manufacturing company, ABC Ltd, produces custom furniture. The company tracks a mixed production overhead that includes a fixed monthly element (such as equipment lease and support) plus a usage-related element linked to output volume (X = units produced).

The following data has been recorded over the past six months:

• January: 1,000 units, £5,000 total cost
• February: 1,200 units, £5,400 total cost
• March: 1,500 units, £6,000 total cost
• April: 1,800 units, £6,600 total cost
• May: 2,000 units, £7,000 total cost
• June: 2,200 units, £7,400 total cost

The company wants an estimated split of fixed and variable cost to improve budgeting and forecasting.

Required

1. Calculate the variable cost per unit (b) using the high–low method.
2. Determine the fixed cost component (a).
3. Construct the cost equation.
4. Forecast the total cost for 1,600 units.
5. Comment on limitations of the high–low method in this scenario.

Solution

1) Variable cost per unit (b)

High activity observation:

X_high = 2,200 units
y_high = £7,400

Low activity observation:

X_low = 1,000 units
y_low = £5,000

Variable cost per unit:

b = (y_high - y_low) / (X_high - X_low)
b = (£7,400 - £5,000) / (2,200 - 1,000)
b = £2,400 / 1,200
b = £2 per unit

So:

b = £2 per unit (of X)

2) Fixed cost per period (a)

Using the high observation:

a = y_high - bX_high
a = £7,400 - (2 × 2,200)
a = £7,400 - 4,400
a = £3,000

So:

a = £3,000 per month

Cross-check using the low observation:

a = y_low - bX_low
a = £5,000 - (2 × 1,000)
a = £3,000

3) Cost equation

y = a + bX
y = £3,000 + 2X

4) Forecast total cost at 1,600 units

y = £3,000 + 2(1,600)
y = £3,000 + £3,200
y = £6,200

So:

Forecast total cost = £6,200

5) Limitations in this scenario

• Two-point sensitivity:only the highest and lowest activity months are used. If either month contains abnormal items, the result can change materially.
• Linearity assumption:the method assumes a stable linear relationship throughout the relevant range.
• Capacity and step effects:if output approaches a capacity limit, costs may rise in steps (extra shift, additional supervision, additional equipment), and the straight-line model may understate costs.

Variant note (judgement trigger)

If one month included a one-off repair or abnormal waste, the recorded cost would be higher than the underlying cost behaviour. Investigate unusual items before relying on the estimate, or use a method that considers all points (for example, a scattergraph and regression).

Interpretation of the results

The estimate suggests:

• a base monthly cost of£3,000(incurred even at low activity within the relevant range), and
• an additional£2for each extra unit produced (X).

A sensible check is to compare predicted and actual costs for a mid-range month. For example:

At X = 1,800 units:
Predicted y = £3,000 + 2(1,800) = £6,600

This matches the April observation in the data set, supporting the linear assumption for this particular set of figures.

Common pitfalls and misunderstandings

• Inverting the high–low formula:b must beΔcost ÷ Δactivity(not Δactivity ÷ Δcost).
• Choosing highest and lowest cost instead of highest and lowest activity:select the high and lowXvalues.
• Not stating units:b must be expressed as “£… per unit of X”.
• Forgetting the cross-check:calculating a from both the high and low observations should give the same answer.
• Extrapolating beyond the relevant range:capacity constraints and step costs can invalidate the model.
• Ignoring abnormal months:unusual repairs, downtime, or temporary pricing changes can distort estimates.
• Using a weak driver:a neat equation is not useful if X does not explain cost movement.

Summary and further reading

Mixed costs include a fixed element (a) and a variable element (bX). The high–low method gives a quick estimate by using the highest and lowest activity observations to fit a straight-line cost model.

Copy/paste-friendly key formulas:

Cost model: y = a + bX
Variable rate: b = (y_high - y_low) / (X_high - X_low)
Fixed cost: a = y - bX (using any observation within the relevant range)

High–low is fast but sensitive to outliers and step changes. When more data exists, a scattergraph and least squares regression typically give stronger estimates.

FAQ

What is the high–low method and when is it used?

It is a two-point technique that estimates a mixed cost’s fixed and variable elements using the highest and lowest activity observations. It is most suitable when a quick planning estimate is needed and cost behaviour appears broadly linear.

How do you calculate the variable cost per unit?

Use change in total cost ÷ change in activity (Δy ÷ ΔX), using the highest and lowest activity observations.

What are the main limitations of the method?

It relies on only two observations, so it is sensitive to outliers and unusual months. It also assumes a linear relationship and may not reflect step changes or capacity constraints.

Why does the relevant range matter?

The estimate is usually reliable only within the normal operating band. Outside that band, extra resources, overtime, or capacity changes may alter cost behaviour.

What should you do if you suspect abnormal data?

Investigate the cause (for example, one-off repairs or abnormal waste). If the abnormal element is not expected to recur, adjust or exclude it before estimating, or use an approach that uses all points such as a scattergraph and regression.

Glossary

Mixed cost
A cost with both a fixed (base) element and a variable element that changes with activity.

Fixed cost per period (a)
The base cost that stays constant in total within the relevant range.

Variable cost per unit (b)
The additional cost for each extra unit of activity (each unit of X).

Cost driver (X)
A measurable activity factor used to explain and predict the behaviour of a cost.

Total cost (y)
The total mixed cost observed for the period at a given activity level.

Cost equation
A linear model used to predict total cost: y = a + bX.

Relevant range
The activity band over which the assumed fixed and variable behaviour is expected to remain valid.

Outlier
An observation that does not fit the general pattern and may distort estimates.

Step cost
A cost that remains fixed over a range of activity but increases to a higher fixed level once capacity is exceeded.