Everyone knows a two-hour marathon is fast.

Almost nobody knows how fast, in numbers you can feel.

This takes an entire marathon field — 31,842 Boston finishers — fits the distribution they actually form, and then marks the world record on it. The record is not near the fast end of this curve. It is past everyone, including the man who won the race, and it sits 2.4 standard deviations below the field mean. The median finisher is 93% slower.

Read the short version or the code and data

3:52

the median of a real marathon field. The distribution is log-normal, mean 4:03, tailing out toward seven hours.

2:00:35

the world record, faster than all 31,842 finishers — the race winner included, at 2:09.

ξ ≈ −0.15

the EVT shape parameter on the fastest 5%. Below zero, so the tail has a finite edge rather than running on forever.

~93%

of the field is beaten by a 3:00 marathon. 4:00 lands almost exactly on the median.

Start · the question

Two questions that are really one

"Where does my time sit?" and "how superhuman is the record?" are usually asked separately, and both get answered with adjectives. Put on the same axis they answer each other: your time is a point on a curve, and the record is another point on the same curve, a very long way along it.

Boston 2014 supplies the curve. It is the largest clean, public marathon field available, with a real distribution rather than a summary.

↓ the curve

A marathon field is log-normal

Fitted, the field is a textbook log-normal: a median of 3:52, a mean pulled later to 4:03 by the slow side, and a long right tail out past seven hours. That asymmetry is not noise, it is what a field looks like — you can be very much slower than the middle, but only so much faster.

Mark the world record on the same axis and it does not sit at the sharp end of the curve. It sits off it.

Distribution of Boston Marathon finish times with a fitted log-normal curve, and a marker for the 2 hour 35 second world record placed far to the left of the entire field. — The whole field as one distribution, with the record marked well outside it.

↓ the elite edge

The fast tail has a wall at the end of it

ξ ≈ −0.15

Peaks-Over-Threshold GPD shape parameter, fitted on the fastest 5%.

A negative shape parameter is the interesting result. It means the tail is bounded: performance does not extend indefinitely into the fast direction but runs into a finite edge. That is consistent with a hard human ceiling, which is what most people assume but rarely test.

What it deliberately does not do is name the ceiling. Peaks-Over-Threshold can identify that a bound exists far more reliably than it can say where the bound is, so the shape is reported and the point estimate is not.

Extreme Value Theory tail fit over the fastest five percent of finishers, showing the fitted generalised Pareto curve bending toward a finite endpoint. — The fitted tail. Negative shape, so it ends rather than continues.

↓ your race

Where any given time lands

The practical output is a lookup. A 3:00 marathon beats about 93% of this field. A 4:00 sits right at the median. Every level's gap to the record is tabulated in you_vs_goat.csv, because the second number is the one that makes the first one mean something.

Percentile curve mapping finish time to the share of the field beaten, rising steeply through the three to four hour range. — Finish time against percentile of the field.

↓ the caveat that matters most

Boston is a qualifying race

This is not the general public. Every runner in this distribution had to post a qualifying time to be here, so the whole curve is shifted fast relative to a typical big-city marathon. When the page says a 3:00 beats 93% of the field, it means this field, and says so.

The direction of the bias is worth noting: a general-entry race would sit slower, which would make the record look more extreme, not less. The caveat cuts against the headline being overstated, not for it.

One number is also withheld on purpose. The record's rarity as a "1 in N" figure would require trusting the fitted density far past where any data supports it, so the record is described empirically instead: faster than everyone, 2.4 SD out.

Finish · how it was built

Where the numbers come from

Data: Boston 2014 finishers via the public llimllib/bostonmarathon mirror, cleaned to runners with plausible times
Sample: 31,842 finishers
Body fit: Log-normal, fitted to the full field
Tail: Peaks-Over-Threshold generalised Pareto on the fastest 5%
Reported: Tail shape ξ, not an estimated fastest-possible time, which POT cannot pin down
Built with: Python, pandas, scipy, matplotlib