Cost behaviour and estimation: high-low, regression and correlation

Budgets, break-even analysis, marginal costing and most decisions need costs split into their fixed and variable parts. But many costs arrive as a single figure: the electricity bill, the maintenance cost, the production overhead total. This lesson shows how to separate them with the equation y = a + bx, the high-low method and least squares regression, and how the correlation coefficient tells you whether to trust the result. ICAN tests high-low often in Section A and can set a full regression and correlation computation in Section B.

Cost segregation and cost estimation

Cost segregation is the separation of a semi-variable (mixed) cost into its fixed and variable elements. Cost estimation is the prediction of a cost at a planned level of activity, usually from those elements. Costs can be segregated by inspecting the accounts, by engineering studies or with a scattergraph; the two methods ICAN requires you to compute are the high-low method and least squares regression.

Linear functions

A straight-line (linear) relationship between cost and activity is written:

y = a + bx

On a graph with activity across and cost up, the line crosses the cost axis at a and rises by b for each unit of activity. If a is zero, the line starts at the origin (a purely variable cost); if b is zero, it is horizontal (a purely fixed cost); if b is negative, it slopes downwards.

The gradient between two points is the change in y divided by the change in x. A haulier's fleet costs ₦1,330,000 at 4,000 kilometres a month and ₦1,570,000 at 6,000 kilometres. Subtracting the equations 1,570,000 = a + 6,000b and 1,330,000 = a + 4,000b gives 240,000 = 2,000b, so b = ₦120 a kilometre and a = 1,330,000 − (4,000 × 120) = ₦850,000. The high-low method is exactly this calculation.

The high-low method

1. Take the highest and lowest activity levels (not the highest and lowest costs) and their total costs.
2. Variable cost per unit, b = (cost at high activity − cost at low activity) ÷ (high activity − low activity).
3. Fixed cost, a = total cost at either level − (b × that activity level).
4. Write y = a + bx and use it to estimate the cost at the activity required.

Two adjustments are common:

• A step in fixed costs. If fixed costs rise by a known amount above an activity level that lies between the low and high points, deduct the step from the high point's cost before step 2, so both points share one fixed-cost basis. When estimating, use the fixed cost that applies at the activity required.
• Inflation. If costs come from periods with different price levels, restate them to one price level with a price index before step 2.

High-low is quick and easy to understand, but it uses only two observations, which may be untypical (a month with a breakdown), ignores the rest of the data, assumes a straight line and gives no measure of reliability.

Least squares regression

Least squares regression uses every observation to find the line of best fit: the line that minimises the sum of the squared vertical distances between the points and the line. Solving its two normal equations (Σy = na + bΣx and Σxy = aΣx + bΣx²) gives:

Here n is the number of pairs of observations and Σ means "the sum of". Lay out a table with columns for x, y, xy and x² (and y² for correlation), total each column, then substitute. Σx² (square each x, then add) is not (Σx)² (add, then square).

Regression uses all the data, but still assumes a straight line, one cause of cost, and past conditions continuing. A forecast within the range of past activity (interpolation) is more reliable than one outside it (extrapolation), where fixed costs may step up.

The correlation coefficient

Correlation is the degree to which two variables are related in a straight line. The correlation coefficient, r (the product-moment, or Pearson's, coefficient) measures it:

r = (nΣxy − ΣxΣy) ÷ √[(nΣx² − (Σx)²) × (nΣy² − (Σy)²)]

The numerator is the same as in the formula for b, so calculate it once and use it twice.

r lies between −1 and +1 (anything outside means an arithmetic error). +1 is perfect positive correlation, every point on a rising straight line; close to +1 is strong positive; close to 0 means little or no linear correlation; close to −1 is strong negative (y falls as x rises); −1 is perfect negative. The coefficient of determination, r², is the proportion of the change in y explained by the change in x. If r = 0.9, r² = 0.81: 81% of the variation in cost is explained by activity, and 19% by other factors.

Correlation does not prove cause: two figures can move together by coincidence, or because a third factor drives both. A high r from four or five observations is weaker evidence than from twenty, and r measures straight-line relationships only.

Worked example

Part 1: high-low with a step in fixed costs. Plateau Cartons Ltd recorded these production overheads. Fixed overheads rise by ₦600,000 a month when output exceeds 7,000 units (an extra supervisor and a leased machine). Estimate the overhead at 6,000 units and at 8,000 units.

Step 1: highest activity is March (9,000 units, ₦5,940,000); lowest is April (4,000 units, ₦3,540,000).

Step 2: March is above 7,000 units and April below, so remove the step: ₦5,940,000 − ₦600,000 = ₦5,340,000.

Step 3: b = (₦5,340,000 − ₦3,540,000) ÷ (9,000 − 4,000) = ₦1,800,000 ÷ 5,000 = ₦360 a unit.

Step 4: fixed cost up to 7,000 units = ₦3,540,000 − (4,000 × ₦360) = ₦3,540,000 − ₦1,440,000 = ₦2,100,000. Above 7,000 units it is ₦2,100,000 + ₦600,000 = ₦2,700,000. Check with March: ₦5,940,000 − (9,000 × ₦360) = ₦2,700,000.

Ignoring the step gives b = ₦2,400,000 ÷ 5,000 = ₦480 and a = ₦1,620,000, and both estimates are wrong (₦4,500,000 and ₦5,460,000).

Part 2: regression and correlation. Makurdi Feed Mills Ltd recorded output (x, in thousands of bags) and production overhead (y, in ₦'000) for six months. Find the regression line, estimate the overhead for a month of 15,000 bags, and calculate and explain the correlation coefficient.

Gradient:

• nΣxy − ΣxΣy = (6 × 67,260) − (90 × 4,140) = 403,560 − 372,600 = 30,960
• nΣx² − (Σx)² = (6 × 1,522) − 90² = 9,132 − 8,100 = 1,032
• b = 30,960 ÷ 1,032 = 30

Intercept: a = (4,140 ÷ 6) − 30 × (90 ÷ 6) = 690 − 450 = 240.

The regression line is y = 240 + 30x. Remembering the units, fixed overhead is ₦240,000 a month and variable overhead is ₦30,000 per thousand bags, or ₦30 a bag. For 15,000 bags: y = 240 + (30 × 15) = 690, so the estimate is ₦690,000, an interpolation within past output of 8,000 to 23,000 bags.

Correlation:

• nΣy² − (Σy)² = (6 × 3,013,800) − 4,140² = 18,082,800 − 17,139,600 = 943,200
• r = 30,960 ÷ √(1,032 × 943,200) = 30,960 ÷ √973,382,400 = 30,960 ÷ 31,199 = 0.99
• r² = 0.9923² = 0.985

There is very strong positive correlation: overhead rises closely in line with output. About 98.5% of the variation in overhead is explained by output, so the line is a reliable basis for budgeting within this range. (High-low on the same data gives b = 455 ÷ 15 = 30.33 and an estimate of about ₦662,000, because it ignores four months.)

Common mistakes

• Picking the months with the highest and lowest cost instead of the highest and lowest activity.
• Forgetting to remove a step in fixed costs before calculating b, or using the wrong fixed cost when estimating.
• Treating Σx² as (Σx)², the most common slip in regression.
• Forgetting the units: if x is in thousands and y in ₦'000, interpret b and a accordingly.
• Saying a high r proves that x causes y, or quoting r² when the question asks for r.

Exam technique

In Section B, set out the table of x, y, xy, x² and y² with totals first: those marks are easy and every later step depends on them. Write each formula, substitute, then answer, so method marks survive an arithmetic slip. Finish with the cost function in words ("₦240,000 a month plus ₦30 a bag") and a sentence on the strength and direction of correlation. In Section A, wrong options usually come from the mistakes above.

Study with AI

I am studying ICAN Management Accounting (Foundation), topic A5: cost estimation by
the high-low method (including a step in fixed costs), least squares regression and
the correlation coefficient. Give me a data set of five or six months for a Nigerian
business in naira and ask me to calculate the cost function, a forecast and r. Wait
for my answer, then show the full working with the table of x, y, xy, x² and y².

AI assistants often slip in regression arithmetic: recompute every total yourself and check the method against these notes.

Lecturer's note: High-low and regression ask the same two questions: how much does the cost rise for each extra unit, and what is left when activity is zero? High-low answers from two points, so one odd month can mislead it; regression answers from every point. The correlation coefficient then tells you how far to trust the answer before you build a budget on it.

Key points

• Cost segregation splits a mixed cost into fixed and variable parts; cost estimation predicts the cost at a planned activity level.
• In y = a + bx, a is the fixed cost (intercept) and b the variable cost per unit (gradient).
• High-low: b = change in cost ÷ change in activity between the highest and lowest activity; remove any step in fixed costs first.
• Regression: b = (nΣxy − ΣxΣy) ÷ (nΣx² − (Σx)²) and a = ȳ − b x̄; it uses all the data and is safest within the range of past activity.
• r, between −1 and +1, measures the strength and direction of a linear relationship; r² is the proportion of variation explained; correlation is not causation.

Quick check

1. A cost function is y = 180,000 + 45x, where y is in ₦ and x in units. Estimate the cost at 6,000 units.
2. A cost was ₦1,050,000 at 3,000 units and ₦1,850,000 at 7,000 units, the lowest and highest activity levels. Find the variable cost per unit and the fixed cost by the high-low method.
3. The correlation coefficient between a product's price and units sold is −0.85. Explain this and calculate r².

Answers

1. ₦180,000 + (45 × 6,000) = ₦180,000 + ₦270,000 = ₦450,000.
2. b = (₦1,850,000 − ₦1,050,000) ÷ (7,000 − 3,000) = ₦800,000 ÷ 4,000 = ₦200 a unit; a = ₦1,050,000 − (3,000 × ₦200) = ₦450,000.
3. Strong negative correlation: as price rises, units sold fall, closely in a straight line. r² = 0.85² = 0.7225, so about 72% of the variation in units sold is explained by price.