The shape decides the answer.

So the first job is to write the range down. The second job, which almost nobody does and which turns out to matter more, is to decide what shape it has.

Three points, and what the words have to mean

The standard move is to replace one number with three. Best case, most likely, worst case. It takes about the same time as arguing about the single number did.

The trap is in the words, and it is a serious one.

“Worst case” does not mean the worst thing you can imagine. It also does not mean the worst you have personally seen. It has to mean something a second person could arrive at independently — a percentile. The figure you would only exceed on one job in twenty.

Without that definition you get what you always get. Ask six engineers for a range they are ninety percent confident contains the final reinforcement tonnage on this job, and you get six comfortable brackets around whatever number is already in their head. On this project the final figure was 228 tonnes. Two of the six ranges contained it.

They were not ninety percent ranges. They were guesses with brackets on.

There is a fix, and it costs nothing. Ask for the extremes first. What would have to go wrong for this to reach its worst? What would have to go right for its best? Only after both of those have been described in words do you ask for the middle. Ask for the most likely first and every subsequent answer is anchored to it — which is exactly the mechanism from Week 7, running inside a single conversation.

Same three numbers, three answers

Now take steel escalation, properly ranged this time.

The best case is that the index is flat or falls, and it costs nothing. The most likely, on the current forecast, is around $18,000. The worst case — a genuine spike of the kind the market produced twice in the last fifteen years — is $84,000.

Those three numbers are now fixed. Watch what happens when you draw different lines between them.

A triangular distribution runs straight lines from the minimum up to the peak and back down. Its mean is simply the three numbers averaged: $34,000.

A PERT curve weights the most likely value four times more heavily and rounds the corners, on the reasoning that the peak is the thing your estimator actually knows about. Its mean is $26,000.

A uniform distribution says every outcome between the bounds is equally likely, because you genuinely have no information about the middle. Its mean is $42,000.

Nothing changed except the shape. The answer moved by sixteen thousand dollars, and the widest is 1.6 times the tightest.

The register, meanwhile, was carrying $30,000 at fifty percent. Fifteen thousand. Every shape says more.

Choosing the shape

Four rules cover almost everything you will meet on a construction project.

Use PERT when you have a real most-likely. If an experienced person can tell you the peak and defend it, that information should be weighted. Gang output, plant durations, most measured work.

Use triangular when the peak is a guess. Triangular is honest about knowing less. It carries more weight in the tails, which is usually the right way to be wrong.

Use uniform when you truly only have bounds — and then go straight back to Week 8, because a uniform distribution is almost always a sign of epistemic uncertainty. It is the shape of not having asked, and the correct response is a phone call rather than a curve.

Use a long right tail for cost. Costs cannot go below zero and they can run a very long way above the mean. A symmetric shape on a cost item is quietly saying that a saving of $40,000 is as likely as an overrun of $40,000, which is not how construction has ever worked.

Events and variabilities are not the same animal

There is one more distinction, and it is the reason a lot of quantitative risk work produces answers that feel wrong.

The crane either fails or it does not. Seventy percent of the time this job runs without it, and the cost of the risk is exactly zero. The other thirty percent, you are somewhere between four and twenty-six thousand dollars depending on when it happens and what it stops. That is an event: a probability, with a distribution sitting behind it, and a large lump of outcomes at zero.

Steel escalation is not like that at all. There is no world in which steel costs nothing to buy. The price will be something. It has no probability of occurring, because it occurs with certainty — the only question is where in the range it lands. That is a variability.

Look back at the register and you will see steel escalation written with a fifty percent probability against it, exactly like the crane.

That is the Week 4 mistake again, one level up. There we found certain costs written as fifty percent risks and only half-funded. Here it is a certain range being discounted by a probability that has no meaning. Steel does not have a fifty percent chance of happening. It has a hundred percent chance of costing you something between nothing and eighty-four thousand dollars.

Practical insight

Take the five largest items on your register and do this in a room with the person who owns each one. Twenty minutes each.

Ask the worst case first, and do not accept a number. Ask what would have to be true for it to get there. If they cannot describe the world in which it happens, the number is decoration.

Then the best case, the same way. Then, last, the middle.

Then ask one final question: if this goes wrong, is it a thing that happens, or a thing that is always happening by some amount? If it is the second one, delete the probability column for that line. It does not have one.

You will find at least one item where the range you get is narrower than the variation you have already seen on site this year. That is the one to go back to.

Key takeaways

✔ A single impact figure is one point lifted from a range nobody wrote down.
✔ “Worst case” must mean a percentile, not the worst thing somebody can imagine.
✔ Six engineers gave six “90% certain” ranges for the steel tonnage. Two of them contained the answer.
✔ Ask for the extremes before the middle, or every answer anchors to the first number said out loud.
✔ The same three numbers give $26,000, $34,000 or $42,000 depending only on the shape drawn between them.
✔ A uniform distribution is usually a sign of epistemic uncertainty — the shape of not having asked.
✔ An event has a probability and a lump of outcomes at zero. A variability always happens. Steel escalation does not have a 50% chance of occurring.

What's coming next

Every one of the fourteen now has a shape instead of a number, and each of those shapes can be sampled thousands of times. That is the machine Track 1 built in Week 15, and it is ready to run.

Except that machine has an assumption baked into it that nobody mentioned at the time, and it is the single largest error in most quantitative risk analysis.

It assumes the bad days arrive independently. That the gang being slow in March has nothing to do with the crane being unavailable in March, and neither has anything to do with the weather that caused both.

Next week: why your simulation is quietly telling you that everything will not go wrong at once, and what happens to the answer when you admit that it will.