The Mean Splits Evenly. The Variance Does Not.

The hypothesis was straightforward. Books don't price a first-half line independently; they derive it from the full-game number with a fixed multiplier — roughly 52% of the total, roughly 55% of the spread. A static multiplier has to be wrong somewhere, and nobody bets these markets hard enough to correct it.

Three ways it could be wrong. I tested all three on 2,639 games, 2016–2025, reconstructed from public play-by-play data.

All three came back negative. And then the fourth thing — the one I only looked at because the first three failed — turned out to be the finding.

Rejected: the share doesn't drift

Ten seasons, and the share never moves more than about a point off 50%. Compare that to the margin distribution over the same period, which drifted materially — seven-point finishes fell 20%. Scoring splits between halves are one of the most stable quantities in the sport. A fixed multiplier is the right model. No staleness edge.

Rejected: the share doesn't depend on the total

This was the main mechanism. If high-scoring games split differently from low-scoring ones, a constant multiplier is systematically wrong at both ends of the board.

Regression of the first-half share on the posted total: slope +0.00065 per point, p = 0.325. No relationship. The bucket-to-bucket wobble isn't even monotonic — it's noise.

Rejected: the spread multiplier is basically right

Empirical fit: first-half margin = 0.559 × full-game spread + 0.47, against a rule of thumb of 0.55. Essentially correct.

The residuals aren't perfectly flat — big favourites over-perform the linear rule in one bucket and under-perform it in the next — but that's non-monotonic across buckets holding 119 and 252 games, with standard errors around ±1 point. Suggestive, not actionable. Underpowered, and I'd rather say so than build on it.

The thing that survived

If two halves were independent with equal variance, the first-half standard deviation would be √0.5 = 0.7071 of the full game's. That's the assumption inside any formula-derived price.

It isn't right. And it misses in opposite directions for two different quantities:

The first half carries 50.4% of the scoring, but 57.3% of the margin variance and only 42.5% of the total variance. The mean splits evenly. The variance doesn't — and it splits the wrong way twice.

The mechanism is clean

Second-half play is partly a response to the first half. A trailing team throws more and stops the clock; a leading team runs it and bleeds time.

That's a mean-reverting force on the margin — which is why there's less margin variance in the second half, and why the first-half-to-second-half margin correlation is −0.027, slightly negative rather than zero. The same behaviour does the opposite to points: comeback attempts and garbage time make second-half scoring more volatile, not less.

One behaviour, two quantities, opposite signs. Which is exactly the signature you'd expect from a single wrong assumption applied to two different things — and it's why the errors are diagnostic rather than random.

The pricing implication: the odds attached to a derived line depend on the variance, not the mean. A model that scales variance by √0.5 understates first-half margin dispersion by about 7% and overstates first-half total dispersion by about 8%. Tails away from the number get priced too cheap in one market and too rich in the other.

Why I'm not trading it

Because establishing what the true distribution is does not establish that anyone is pricing it wrong. Those are two different claims, and I've only tested the first.

Settling the second needs a historical archive of posted derived lines, which is the one thing no cheap data source carries. Without it, I have a well-measured property of the world and no evidence about anyone's model. Those get written down differently.

What generalises

Square-root-of-time scaling is an assumption, not an identity. It follows from independent, identically distributed increments. Wherever the increments respond to the accumulated path — and in markets they very often do — it breaks, and the direction of the break depends on whether the feedback is stabilising or destabilising.

Volatility that mean-reverts within the period scales slower than √t. Trending or momentum-driven volatility scales faster. Intraday volatility patterns are the obvious equity case: a U-shaped intraday profile means the volatility of a partial session is not √(fraction) of the full session's, and any option or risk calculation that assumes otherwise is wrong in a knowable direction.

And the meta-lesson, which is the one I'll actually remember: I went in expecting the means to be wrong and they were right. The variance was the thing worth measuring, and I would not have looked at it if the first three tests had come back positive.

Rejected hypotheses aren't wasted work. They're what makes you look somewhere you'd otherwise have walked past.