If you are summing the rolls of three dice, how does the expected value and variance change if you add one dice and only sum the highest three?

The problem is:

An alternative method to generating your character is by rolling four six sided dice and summing only the three highest. What is the expected value and variance of this procedure?

The previous question had the same calculation but by simply rolling 3 dice instead of 4 and summing all of them. What effect does adding the fourth dice have? Does is raise the expected value?

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Number of cases  : 1296
Mean of score    : 12.2445987654
Variance of score: 8.1045232958


This was computed by the following Python program:

values = [1, 2, 3, 4, 5, 6]

s0 = 0
s1 = 0
s2 = 0

for a in values:
for b in values:
for c in values:
for d in values:
throws = [a, b, c, d]
throws.sort()
throws = throws[1:]
score = sum(throws)
s0 += 1
s1 += score
s2 += score*score

mean = float(s1)/s0
variance = float(s2)/s0 - mean*mean

print "Number of cases  :", s0
print "Mean of score    :", mean
print "Variance of score:", variance


The answer is: yes, it raises the mean (from $10.5$), but it reduces the variance (from $8.75$).

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Do you know of any way to arrive at this answer without using a computer? –  eatinasandwich Nov 25 '13 at 3:17
It looks like a hard problem. I can't see a way to solve it by hand. Maybe there is one. –  apt1002 Nov 25 '13 at 6:23
@eatinasandwich there is not a good formula (such as the counting formula for combinations and permutations etc.) for this problem because the sort ordering required. It comes down to simply counting all possibilities, this answer is probably as good as it gets. –  adam W Nov 25 '13 at 23:19