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I use it in biostatistics. For example, why do genes (or any input value) with a larger absolute fold-change also tend to have a more significant p-Value? All volcano plots look like it, so I assume, it's a mathematical reason?

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The p-values come from the student's t-test, which is itself a function of the change in the mean: $$t=\frac{Z}{s/\sqrt{n}}$$ In the volcano plot you have fold changes instead of the mean, but the idea is the same. Since the standard deviation mostly is noise, it must be independet of the real data. So larger fold changes will result in higher t values and subsequently in smaller p values (hence the shape).

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