Linear Weights, Explained: Why a Double Isn't Worth Two Singles

Ask someone how much more a double is worth than a single and the instinctive answer is “twice as much — it’s two bases instead of one.” It’s a clean answer, it’s intuitive, and it’s wrong. A double is worth a lot more than a single, but nowhere near twice as much, and the gap between intuition and reality is the whole reason linear weights exist. They are the bridge between “a double has two bases” and “a double produces about this many runs,” and once you see how that bridge is built, a surprising amount of modern baseball analysis stops looking like magic.

The short version: a linear weight is the average number of runs an event adds, measured from real games, not assigned by anyone’s sense of fairness. Every walk, single, double, triple, and home run gets one. String them together and you get the run value of a hitter’s whole season — and, as it happens, the machinery underneath wOBA.

Why a double isn’t worth two singles

Start with the intuition and watch it break. A single moves the batter to first and typically nudges existing runners up a base. A double moves the batter to second and tends to push runners two bases — often scoring a man from first, almost always scoring one from second. So a double clearly does more than a single. The question is how much more, and the answer is not “double it.”

The reason is that the first base is the most valuable one. Just getting on base — turning an out into a baserunner — is the bulk of the value, because outs are the scarce resource that ends innings. The extra base a double buys on top of that is real but incremental; it advances runners further and puts the batter in better scoring position, but it doesn’t double the “not making an out” part, since the single already banked that. So in run terms a double comes out worth roughly 1.4 times a single, not 2 times. A triple isn’t three singles either, and a home run — which clears the bases and scores the batter guaranteed — lands at something like two and a half times a single, not four. Bases and runs are different currencies, and linear weights are the exchange rate.

How run values are actually derived

Here is where it gets satisfying, because the weights aren’t guessed — they fall straight out of run expectancy. Recall that every moment of an inning is a base-out state: which bases are occupied and how many outs there are, twenty-four states in all, and each has a measured expected-runs value computed from years of play-by-play.

To value an event, you do one subtraction. Find every time a double happened, look at the run expectancy of the state right before it and the state right after, take the difference, and add any runs that scored on the play. Average that over thousands of doubles in every base-out situation, and you get the run value of a double — the average amount it improved the offense’s run outlook.

Do that for each event type and the linear weights assemble themselves. A representative recent-season set of run values, expressed as runs above a generic out, looks roughly like this — treat them as representative, not as any specific year’s official figures, since the exact decimals drift with the run environment:

Look at the single-to-double relationship in those numbers: about 0.45 versus about 0.76. That’s a ratio near 1.4, exactly the “not twice as much” from earlier, now falling out of the run-expectancy arithmetic instead of from anyone’s opinion. And notice the out carries a negative weight — making an out actively costs runs, which is why on-base skill matters so much. The event values are real and positive; the out is the tax every plate appearance risks paying.

From run values to wOBA

Now the payoff, and the reason this article exists. wOBA is, almost exactly, linear weights with the numbers rescaled to look like a stat you already read. Take the raw run values, shift and stretch them so the league average lands on the on-base-percentage scale — roughly .320 — and you get the familiar wOBA coefficients. The relative spacing is preserved; only the units change. A double is still worth about 1.4 singles inside wOBA, because it was worth about 1.4 singles in run value, because run expectancy said so.

That’s the entire trick. wOBA isn’t a separate theory of hitting — it’s linear weights wearing an on-base-percentage costume so the number sits in a range your eye already calibrates. When you read “.370 wOBA is very good,” you’re reading run values that have been politely rescaled. Here is the standard form, with a representative recent-season weight set:

Those coefficients are the linear weights, rescaled. The 1B-to-2B step from 0.89 to 1.27 is that same 1.4-ish ratio one more time, and the home run at 2.10 is roughly two and a half singles — not four. The philosophy is identical to the raw run values; only the dressing changed. If you want to watch it work, plug a line into the calculator below and then read the weights left to right: each coefficient is just the run value of that event, scaled to the on-base world.

Type a real or hypothetical batting line into that and you’re running the linear-weights engine by hand. The reason a walk-and-power hitter can post a great wOBA on a mediocre batting average is sitting right there in the coefficients: walks and homers carry their full run value, and batting average never sees the walk at all.

Why linear weights, and where they bend

The word “linear” is doing real work and deserves a caveat. The method assumes each event contributes a roughly fixed average run value regardless of context — it deliberately ignores whether the double came with the bases loaded or empty. That’s a feature when you want to measure a hitter’s context-neutral skill, because it strips out the luck of when his hits happened, which isn’t a repeatable talent. It’s a limitation if you want to credit what actually occurred in specific situations — that’s the job of RE24, which values each play by its real base-out change rather than a season-average weight.

So the two live side by side and answer different questions. Linear weights and wOBA ask “how good is this hitter, on average, per the run value of what he does?” RE24 asks “how many runs did this hitter’s actual plate appearances, in their actual situations, produce?” Same run-expectancy foundation, different question about context.

The bottom line

A double isn’t worth two singles because bases aren’t runs, and linear weights are the measured exchange rate between them. Derive each event’s run value from the change it makes in run expectancy, and you get a set of weights that rank everything a hitter can do by how much it actually helps a team score. Rescale those weights onto the on-base scale and you’ve built wOBA. Every time you read a wOBA, you’re reading run expectancy in disguise — and now you know exactly what’s under the costume.