= Common-coin averaging
In the copied-coin model, $X_i=Z$ for every $i$, so the <sample mean> is exactly $Z$, however large $n$ is. Its <expected value> is $1/2$ and its <variance> is $1/4$, matching each observation's marginal variance. The off-diagonal <covariances> in the <variance of a sum> explain the difference:
$$
\operatorname{Var}(\overline X)=\frac1{n^2}\sum_{i,j=1}^n\operatorname{Cov}(X_i,X_j)=\frac1{n^2}\sum_{i,j=1}^n\frac14=\frac14.
$$
For an <IID sample>, those off-diagonal terms are zero. Repeating the same random input therefore does not create the reduction in fluctuations obtained from fresh independent draws. The general common-plus-individual structure is described by <compound-symmetry covariance>.
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