  We consider a one-dimensional stochastic differential equation driven by a
Wiener process, where the diffusion coefficient depends on an ergodic fast
process. The averaging principle is satisfied: it is well-known that the slow
component converges in distribution to the solution of an averaged equation,
with generator determined by averaging the square of the diffusion coefficient.
  We propose a version of the averaging principle, where the solution is
interpreted as the sum of two terms: one depending on the average of the
diffusion coefficient, the other giving fluctuations around that average. Both
the average and fluctuation terms contribute to the limit, which illustrates
why it is required to average the square of the diffusion coefficient to find
the limit behavior.
