PHY.8:4.4 - Establish when the aggregate represents a typical observation

Compare the predicted spread with the tolerance or decision in the work. For any scalar observable Y with finite variance and positive tolerance epsilon, Chebyshev’s inequality gives

P(|Y-E[Y]| >= epsilon) <= Var(Y)/epsilon².

This can settle a sufficient bound without reconstructing the whole distribution. If the bound is too loose to decide, a sharper calculation may help. Its cost is justified by the unresolved decision, not by the mere availability of another statistical method.

For an average over N constituents, the variance is bounded by a constant times 1/N when the sum of relevant covariances is bounded above by a constant times N. A common fluctuating influence can instead make the total variance grow as N². Inspect the physical dependence that determines this scaling. Increasing the number of constituents then has different effects on reliability.

Keep a probability claim relative to its measure. A set containing most of the probability need not contain most of the unweighted alternatives. Conversely, a large count of states says little about the prepared distribution until its weights are supplied.

Decide whether a finite system and the requested observable permit the limiting argument. Correlation lengths comparable to system size, constraints, long-range interactions or operation near a transition can invalidate the approximation used to obtain concentration. Return to the physical account when that invalidation changes the required result; do not require a thermodynamic limit for an already sufficient finite calculation.