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Variance of averaged measurements of unequal size
Suppose that we have a random variable X that we can only take averaged samples from. This means we cannot directly obtain a sample Xi, but only a sample of the mean of k Xi's. To complicate things, k varies uncontrollably with each measurement. How do we estimate the variance of the original variable Var(X)?
Call the averaged samples we obtain Ski, where ki denotes the number of X's that have been averaged. Then we have for a fixed value of ki (see e.g. this):
Var(Ski)=⟨(Ski−⟨Ski⟩)2⟩=⟨(Ski−⟨X⟩)2⟩=Var(X)ki,
where ⟨⟩ indicates the expectation value and we somehow know ⟨X⟩, to avoid having to care about Bessel's correction. More generally, we have for all ki ⟨ki(Ski−⟨X⟩)2⟩=Var(X).
If we have N random averaged samples Sk1, Sk2, …, SkN, then ⟨k1(Sk1−⟨X⟩)2+k2(Sk2−⟨X⟩)2+...+kN(SkN−⟨X⟩)2⟩=⟨k1(Sk1−⟨X⟩)2⟩+⟨k2(Sk2−⟨X⟩)2⟩+…+⟨kN(SkN−⟨X⟩)2⟩=Var(X)+Var(X)+…+Var(X)=N⋅Var(X). In other words, the expectation value of every term ki(Ski−⟨X⟩)2 is Var(X), so we can estimate Var(X) by computing 1NN∑i=1ki(Ski−⟨X⟩)2.