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Equation 4 · Part 12 · Measuring Frontier Models: Contamination, Variance, and What a Score Can Support

Ending index or upper bound: n

s^=1n∑i=1n1 ⁣[ success on xi ],xi∼Dbench.\hat{s} = \frac{1}{n}\sum_{i=1}^{n} \mathbf{1}\!\left[\,\text{success on } x_i\,\right], \qquad x_i \sim \mathcal{D}_{\mathrm{bench}} .
nn

What this part means

This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.

Its job in the formula

n occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

Reframe the object. A benchmark score s^\hat{s} is an estimator of a population quantity s — the model’s success rate over some distribution of tasks D\mathcal{D} that somebody hopes resembles the work you actually have. Every property that makes an estimator trustworthy applies: s^=1n∑i=1n1 ⁣[ success on xi ],xi∼Dbench\hat{s} = \frac{1}{n}\sum_{i=1}^{n} \mathbf{1}\!\left[\,\text{success on } x_i\,\right], \qquad x_i \sim \mathcal{D}_{\mathrm{bench}} . The number is only as good as three things: whether Dbench\mathcal{D}_{\mathrm{bench}} resembles D\mathcal{D} , whether the xix_i are genuinely held out, and whether the indicator is measured with enough repetition to characterise its spread. All three fail routinely, and they fail in different directions.

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Learn the underlying idea

Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.

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