Hull watched rats run a maze and found they ran the last stretch faster than the first. The version that reaches design is the progress bar, the step counter, the five-stamp loyalty card, and the assumption that showing someone how far along they are will make them finish.
That assumption is testable, and testing it properly means dealing with the thing that makes it hard: you get faster at anything as you repeat it. A run that speeds up toward the end has two explanations, and practice is the boring one. Any demonstration that only shows you accelerating has shown you nothing.
So there are two runs below, and they are the same fifteen steps in the same order — same targets, same positions, same everything. In one of them a progress bar sits above the grid and tells you exactly how close the end is. In the other there is no bar, no count, and no way to know when it stops. Practice happens in both. Only one of them lets you see the finish line.
Which run comes first is decided by a coin, so an order effect cannot masquerade as the result.
What the two runs are for
Both runs get faster. That is practice, and it is not interesting. What the page compares is the shape of the speeding-up: how much quicker the last third of a run was than its first third, in the run with a bar against the run without one.
Practice is present in both and cancels. Whatever extra acceleration shows up in the run with the visible finish line is the part that had nothing to do with getting better at the task. If the two shapes come out the same, the honest answer is that this run found no gradient, and the page says that rather than pointing at the raw speed-up and claiming a victory.
Fifteen steps split into thirds is five measurements per point. That is not much, and the verdict below refuses to interpret differences smaller than the noise such a sample carries.
What it costs you
A progress indicator is not decoration, it is fuel — but only if it is believed. The effect depends on the end being visible and credible. A bar that jumps, stalls, or lies about what is left removes the thing that was doing the work.
Endowed progress is the cheapest version of this. A ten-step task shown as twelve steps with two already done is the same ten steps, and it gets finished more often. The head start costs nothing and moves someone out of the flat part of the curve before they have started.
The flat part is the dangerous part. Motivation is lowest in the middle, furthest from both the start and the end — which is exactly where most multi-step flows put their hardest question. Front-load the friction or save it for the end, but do not bury it at step five of nine.
An unknown length is the worst case. Not because the work is longer, but because there is no finish line to accelerate toward, which is also why "loading" with no bar feels so much worse than "loading" with one.