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Say I make 3 independent experiments and these are the outputs:

O/P of $1$st exp : $1,2,3$
O/P of $2$nd exp : $4,5,6$
O/P of $3$rd exp : $7,8,9$

In general ${\rm Var}(A+B+C) = {\rm Var}( A ) + {\rm Var}( B ) + {\rm Var}( C )$

In this case ${\rm Var}(\text{1st}) + {\rm Var}(\text{2nd}) + {\rm Var}(\text{3rd}) \ne {\rm Var}(1,2,3,4,5,6,7,8,9)$

Why ?

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    Informally, the set $\{1,2,\dots,9\}$ has more "wiggle" about its mean. The sum of the means of $A$, $B$, $C$ is $15$, maximum value is $18$, for a maximum wiggle of $3$ about the mean. For the numbers $1$ to $9$, the mean is $5$, maximum wiggle is $4$. Of course maximum wiggle doesn't tell us variance, but it gives some indication as to why the second variance might be larger. This is particularly the case because we are squaring the deviations from the mean, and $4^2$ is a lot bigger than $3^2$.2012-10-08

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