General

Where a Cost Curve Comes From

Every estimator eventually asks where a parametric cost number actually comes from. This article takes a cost curve apart: how a sample of real projects becomes a scatter plot, why the relationship straightens on logarithmic axes, how the regression is fitted, what the exponent and the coefficient each control, and why the dispersion around the fit is information rather than noise.

Carlos Fuenmayor

Cost Engineer

· 19 Jul 2026 · 5 min read · 122 views

Back to Insights

Explanation AI Generated

Every estimator who is handed a parametric cost model asks the same question sooner or later, and it is the right question to ask: where does this number actually come from? If the answer is a shrug, the estimate is worthless — you cannot defend what you do not understand. So this article takes a cost curve apart and shows what is inside it.

Nothing here is specific to any one platform or dataset. This is the method itself, the one described in the cost engineering literature and taught in AACE practice. The figures below are computed from illustrative data generated for this article, so you can follow the arithmetic without any proprietary material getting in the way.

1. It starts with a sample, not with a formula

A cost curve is not derived from theory. It is derived from observations — real projects, real equipment purchases, real installed costs, each one recorded together with the physical quantity that drove it. That physical quantity is the cost driver: barrels per day for a process train, megawatts for a power plant, square metres for a building, kilometres for a pipeline.

Plot them. The driver goes on the horizontal axis, the cost on the vertical axis. Each project becomes one point. What you get is a cloud, and the cloud is the raw material of everything that follows.

The first thing that matters is how many points you have. With a handful, a great many different curves would pass through them equally well, and you have no basis for choosing between those curves. As the sample grows, the possible curves collapse towards one another and the relationship becomes visible. This is what "confidence" means in practice — not a feeling, but the degree to which the data constrains the answer.

Figure 1. The same underlying relationship sampled twice. On the left, six observations: the fitted line is little more than a guess, and a different six points would give a noticeably different line. On the right, forty-five: the slope and the level are now pinned down by the data itself.

2. Why the relationship is not a straight line

Plot capital cost against capacity on ordinary linear axes and you get a curve that bends downward. This is not a statistical artefact; it is engineering. A vessel twice the volume does not need twice the steel, because volume grows as the cube of a dimension while surface area grows as the square. A larger compressor does not need two operators instead of one. Fixed scope — control rooms, roads, permits — is shared over more output. This is economy of scale, and it is the dominant feature of capital cost data.

Figure 2. On linear axes the relationship is a curve. It is difficult to fit by eye, difficult to extrapolate, and difficult to compare between facility types.

Now plot exactly the same points with logarithmic axes on both scales. The curve straightens.

Figure 3. The same data on log–log axes. The relationship becomes a straight line, and the slope of that line is the scaling exponent. This is the reason the power form is the natural mathematical description of capital cost — not because it is convenient, but because that is how the data behaves.

If a relationship is a straight line on log–log axes, it is a power law. That single observation is the bridge between a cloud of project data and an equation you can use.

3. The regression

Because the log–log plot is linear, fitting the curve reduces to fitting a straight line — the oldest and best-understood problem in statistics. Take the logarithm of both the driver and the cost, run a least-squares regression, and you obtain a slope and an intercept. Transform back, and you have the curve.

This is the same operation a spreadsheet performs when you add a power trendline to a scatter chart. There is no magic in it, and that is precisely the point: the procedure is transparent, reproducible and checkable by anyone with the same data.

Figure 4. The fitted relationship through the observations. The regression finds the line that minimises the squared deviations in logarithmic space, which is equivalent to minimising proportional rather than absolute error — appropriate when costs span orders of magnitude.

A check worth doing. The data in these figures was generated from a known exponent of 0.680, then had random scatter added to it. The regression, seeing only the scattered points, recovered an exponent of 0.689. That is what a working regression looks like: it reconstructs the underlying behaviour from noisy observations, and it tells you how confident to be about the reconstruction.

4. Anatomy of the equation

The fitted relationship has the form y = a · xk, and it has exactly three parts. Understanding what each one does is what separates using a cost model from merely operating one.

x is the driver. The physical quantity you supply — the capacity, the length, the area, the throughput. It is the only part you enter, and choosing the right driver matters more than any refinement that comes after it.

k is the exponent, and it governs the shape. It is the slope of the straight line on the log–log plot. It answers the question: when the plant gets bigger, how does the cost respond?

Figure 5. The exponent alone decides the shape. Below 1.0 the cost grows more slowly than the size — economy of scale. At exactly 1.0 cost is proportional to size. Above 1.0 each additional unit of capacity costs more than the last, which happens when scale brings new problems rather than savings.

For process plant equipment the exponent typically sits well below one, which is the quantitative expression of the old rule that bigger is cheaper per unit. For linear infrastructure — a road, a pipeline, a transmission line — it approaches one, because a second kilometre costs very much what the first one cost. Reading the exponent tells you something real about the physics of what you are building.

a is the coefficient, and it governs the level. It does not change the shape at all; it lifts the whole curve up or sets it down. Everything that shifts the cost of a project without changing how it scales lives in a: the country you build in, the year you build in, the market at the time of award, the scope included in the price.

Figure 6. The coefficient shifts the curve vertically without altering its shape. This is where location factors, escalation indices and scope differences act.

A single sentence worth remembering: the exponent describes the engineering, the coefficient describes the circumstances. Confusing the two is the most common error in parametric estimating.

5. The scatter is information, not noise

The points never sit exactly on the fitted line, and a novice reads that as a defect. It is not. The dispersion is telling you something true: projects of the same size genuinely do cost different amounts, because they differ in ways the driver does not capture — different sites, different contracting strategies, different execution quality, different luck.

That dispersion is what a percentile band expresses. The central line is the typical outcome; the lower and upper bounds describe how far the observed projects spread around it. When an estimating tool offers you a P10, a P50 and a P90, this is where those numbers come from.

Figure 7. The dispersion of the observations around the fitted relationship, expressed as a band. The fit is the central expectation; the band is the honest statement of how much the real projects varied around it.

Two cautions belong here, and they matter more than anything else in this article. First, a band of this kind describes the uncertainty of the curve, which is not the same thing as the uncertainty of your project. Second, summing the low ends of many accounts does not produce a genuine low case for the total — that would require a probabilistic simulation of correlated risks, which is a different and heavier exercise. Present a percentile band for what it is and it will withstand scrutiny; present it as a forecast probability and it will not.

6. The range is part of the answer

Every curve was fitted over a specific span of the driver, because that is the span the sample covered. Inside that span you are interpolating between observations, which is what regression is for. Outside it you are asserting that a relationship observed in one size range continues to hold in another, and that assertion may or may not be true.

Figure 8. Inside the observed span the fit is supported by data. Beyond it — shown dashed — the curve is an extrapolation. It may still be the best available answer, but it is a different kind of claim and should be labelled as one in the basis of estimate.

Extrapolation is not forbidden. It is simply a statement that must be made out loud. A great many estimating disputes come down to somebody using a relationship outside the range it was built for and nobody noticing.

7. When one driver is not enough

Capacity explains most of the variation in capital cost, but never all of it. Two transmission lines of identical length differ by voltage, by conductor type and by the terrain they cross. Two refineries of identical throughput differ by the crude they process and the metallurgy that requires. Two offshore platforms of identical production differ by water depth and by the sea they sit in.

There are two ways to handle this. One is to fit a separate curve for every combination, which multiplies the data requirement beyond what is available. The other — the practical one — is to keep a validated reference curve and apply explicit, documented adjustment factors for the conditions that differ from the reference case. The reference configuration applies no adjustment at all, so the underlying curve is never obscured, and every departure from it is visible and defensible.

The discipline this demands is knowing what the curve already contains. If the sample was drawn from one region, the regional effect is already inside the coefficient and must not be applied again. If the driver you entered already embeds a condition, that condition must not be charged twice. Double counting is easy to do, invisible once done, and systematically inflates an estimate.

8. What to take from this

A parametric cost model is a piece of engineering evidence, assembled by a procedure you can follow end to end: collect observations, plot them against the driver that governs them, recognise the power relationship in the log–log linearity, fit it, read the exponent as physics and the coefficient as circumstance, and carry the dispersion forward honestly as a band rather than discarding it.

None of it is magic, and none of it should be taken on faith. If you are handed a curve, the questions to ask are always the same four: what data was it fitted to, over what range, what scope does the price include, and what is the reference case the adjustments are measured from? An estimating tool that cannot answer those is asking you to trust it. One that can is giving you something you can defend in front of a reviewer — which, in the end, is the only kind of estimate worth producing.

A note on what is public and what is not. The method described here is common property of the cost engineering profession, and it should be taught openly — an estimator who does not understand it cannot properly use it. The fitted coefficients of a particular curve library are a different matter: they are the accumulated result of data collection and calibration work, and organisations reasonably keep them internal. Understanding the method does not require access to anyone's dataset, and this article deliberately uses none.

Meet Kpex

Kpex gives project cost estimators instant access to reliable, benchmarked cost data for all major process plant components. With structured models covering equipment, piping, electrical, civil works and more, it delivers faster, more accurate, and defensible estimates — transforming cost data into clear insights that save time, reduce risk, and keep projects competitive.

CREATE FREE ACCOUNT MORE INFORMATION