Great explanation! Just a nitpick: If you don't need the value of $sum outside the loop, declare it inside:

my $ssum = 0;                     # $sum not needed here
for my $i (1..1000){
    my $sum = 0;                  # but here.
    for my $j (1..$sampleSize){
        $sum+=rand();
    }
    $sum/=$sampleSize;
    $ssum+=$sum;
}
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