The single-parameter scaling hypothesis relating the average and variance of the logarithm of the conductance is a pillar of the theory of electronic transport. We use a maximum-entropy ansatz to explore the logarithm of the energy density, ln W(x), at a depth x into a random one-dimensional system. Single-parameter scaling would be the special case in which x = L (the system length). We find the result, confirmed in microwave measurements and computer simulations, that the average of ln W(x) is independent of L and equal to −x/ℓ, with ℓ the mean free path. At the beginning of the sample, var[ln W(x)] rises linearly with x and is also independent of L, with a sublinear increase near the sample output. At x = L we find a correction to the value of var[ln T ] predicted by single-parameter scaling.

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