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I wonder if they can combine the experimental results from the 2.5 sigma and the 3.5 sigma experiments. If two independent experiments confirmed some hypothesis at 2.5 and 3.5 sigma, the combined evidence is surely much larger than 3.5. Does anyone know what the right way of combining experimental results is, to yield a more accurate answer?

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Assuming the errors are random and uncorrelated, the total significance is the square root of the sum of the squares. In this case, that's only 4.3 sigma.

> only 4.3 sigma

I generally treat particle physics as a black box from which magic appears, but my understanding was that 3 is where things start getting interesting, and 5 is where it's considered pretty conclusive, so wouldn't going from 3.5 to 4.3 be a significant jump?

Really depends on the experimental approach, but there are certainly techniques for combining independent results. For example, in genetics, inverse-variance weighted meta analysis is common. Studies are essentially combined with their effect weighted by the degree to which their results are tightly clustered around that study's mean result.

I would be interested to learn about common procedures from physics as well.

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