Assuming you have version 9 you can do the following.
data = {{-1, 0}, {0, 0}, {1, 0}, {-2, 1}, {2, 1}, {-1, 3}, {1, 3}};
dist = EmpiricalDistribution[data];
Table[Expectation[y \[Conditioned] x == i, {x, y} \[Distributed] dist], {i, -2, 2}]
(*{1, 3/2, 0, 3/2, 1}*)
Note: Conditional probabilities and expectations didn't work for EmpiricalDistribution in version 8. In that case you could code this up yourself as...
Table[Mean[Cases[data, {i, y_} :> y]], {i, -2, 2}]
(* {1, 3/2, 0, 3/2, 1} *)
Edit: An incredibly inefficient but distribution-based solution in M8 is to use ProbabilityDistribution.
pdist =
ProbabilityDistribution[Block[{tally = Tally[data], nprobs},
nprobs = Normalize[tally[[All, 2]], Total];
Piecewise[
Transpose[{nprobs, (And @@ Thread[{x, y} == #]) & /@
tally[[All, 1]]}]]
], {x, -3, 3, 1}, {y, -3, 3, 1}]
Table[
Expectation[
y \[Conditioned] x == i, {x, y} \[Distributed] pdist], {i, -2, 2}]
(* {1, 3/2, 0, 3/2, 1} *)
For completeness, to compute the variance using Expectation it gets a little messy.
mu = Expectation[y \[Conditioned] x == i, {x, y} \[Distributed] pdist];
Table[Expectation[(y - mu)^2 \[Conditioned]
x == i, {x, y} \[Distributed] pdist], {i, -2, 2}]
(*{0, 9/4, 0, 9/4, 0}*)