(This article had help from Clyde Kruskal, Joe Kruskal, Bill Kahn, Hans Courant, and Lucy Moser.)
Bill James, the baseball statistician, once had
an article on measuring
Baseball Families.
Who was the best baseball family?
Would it be ...

The Alous: Felipe, Jesus, Matty
were brothers. Moses was Felipe's son.
All made the major leagues, and some of them were
pretty good.

The Bonds: Bobby and Barry Bonds. Father/Son both excellent.
While you might rather have them then the four Alou's
on your team, having four in a family just seems like
a stronger family.

The Aarons: Hank and Tommie Aaron. Hank was a great player
who hit 755 home runs (without steroids).
Tommie, his brother, had a very short career.
Even though its a real family, still doesn't seem like
the right answer.

Babe Ruth alone: Bill James him rates as the best player
of all time. Hence the "Ruth family" would seem to be a
very good baseball family.
So, how to go about the question of
who is the best baseball family ever?
First we need a way to assign to a player
some number saying how good he is.
Bill James has already done this via a method
called
win shares.
The idea is that you assign to a player how much he
helped the team win. This is very complicated and I won't
explain it here. However I will point out that it can't
be negative and it varies quite a bit.
For example Hank Aaron had 643 win shares, while
his brother Tommie had 15.
For some examples of players with high win shares
see
this.
So, you could just take the sum of the family members
win shares. But this makes the
family of one, the winner. We want a real family
to get some credit for being a real family.
Bill James' solution:
Say the best person has a
_{1} win shares, the second
a
_{2}, etc.
a
_{1} > a
_{2} > ... > a
_{n}.
Rate the family via
a
_{1} + 2a
_{2} + ... + na
_{n}.
Under this metric the Alou brothers were the
best baseball family as of 2003.
See
here
for the numbers.
But I am still bothered.
Why the combination
a
_{1} + 2a
_{2} + ... + na
_{n}?
Is there some other way that is more mathematically sound
or that can be derived? I doubt it since the question is
somewhat subjective.
Clyde Kruskal has brought up another point.
What if you had a longer link between relatives?
What if you had a greatgrandfather, grandfather,
father, son. If the father is the best baseball player,
start there. People who share half his genes
(son and grandfather) count fully. But
the greatgrandfather counts less.
You could even do this if someone in the family
does not play baseball.
We will see an example below under
(what else) the Kruskal Math Family.
Who are the best
Math families of all time? Here are some,
not in order. I am sure there are more. Corrections
and additions welcome! (I may make a website out of it.)
I only count a family if it has at least three members and
all of the people are related by blood (sorry Blums).
Aside from that, I am fairly informal. This list is not
meant to be the final word.

The Bernoulli Bunch
Link.
There were eight of them. JakobBernoulli numbers,
NickolausProb theory and Geometry (NOT Bernoulli Dist),
Johann Brachistochrone problem and possibly L'Hopital's rule,
DanielBernoulli's principal (also physics and probability),
Wikipedia does not have info on the other four, but says
they were math folks.

The Kruskal Kin:
William (KruskalWallace test in statistics),
Martin (Solitons),
and
Joseph (Min Spanning tree, Kruskal Tree Theorem (set of all trees under embedding is a well quasi order), KruskalKatona Theorem).
They are all
brothers. Martin's son is
Clyde (Parallel Computation, Coloring the Plane).
Clyde's son is Justin (Ramsey Theory, though he's still in High School, so perhaps shouldn't count).
William's son is Vincent (Computer ScienceIBM research).
Rosaly and Molly Kruskal are sisters of Martin/William/Joseph. They do
not do math, but Molly has two sons who do math: William Kahn (statistician),
and Ted Kahn (Statistical Software); and
Rosalie's son is Jeremy Evnine (OR, Math Finance).
(Using Clyde's theory these people would count some.)
The following does not count as he is not a blood relative:
Joe Kruskal's soninlaw is
Neal Madras
(Math prof at York Univ in Canada).

A Nest of Noethers:
Father:
Max Noether
one of the finest mathematicians of the nineteenth century
according to Auguste Dick who wrote a book on
Emmy Noether.
Max's son was Fritz Noether.
Fritz Noether.
Max's daughter was
Emmy Noether
the most important woman in the history of mathematics according to Einstein and others.
(That was meant as a compliment but sounds so odd nowadays.)
A great mathematician independent of her gender.
The only fathersondaughter combination that I know of.

The Markov Chain:
Andrey Markov
(Markov Chains), his brother
Vladmir Markov
(Markov's inequality coauthored with his brother),
and
Andrey Markov Jr. (logic)
son of Andrey Markov.

The Browder Brothers:
Felix (PDE's),
William (Topology and Geometry),
and
Andrew Browder (Analysis).

A Litter of Lenstras:
Hendrik (Computational Number Theory),
Arjen (Crypto), and
Jan Karel.
Interesting misses:

A Research of Rabin's:
Michael
and
Tal. A FatherDaughter both in Crypto. Probably the only such.

A Manifold of Millars:
Terry and
Jessica. A FatherDaughter both in Recursive Model Theory. Definitely the only such.

A Troop of Tardos':
Eva and
Gabor.
The only other BrotherSister combo I know of is
the Noethers.

The Courant Clan:
Richard Courant (Math),
Hans Courant (son of Richard, Physics),
Ernst Courant (son of Richard, Physics)
Ted Courant (son of Hans, Math),
Jurgen Moser (soninlaw of Richard, Math)
Lucy MoserJauslin (Jurgen Moser's daughter, Math),
Jerry Berkowitz (soninlaw of Richard, Math),
Peter Lax (soninlaw of Richard, Math),
Carl Runge (Fatherinlaw of Richard, Physics and Numerical Analysis).
Note that Jurgen and Lucy Moser are a fatherdaughter combination.

A Band of Blums:
Manuel and
Lenore
(married)
and
their son
Avrim.
All do TCS at CMU.
LenoreAvrim is the only motherson combo I know.

A Geek of Gauss' or A Nerd of Newtons or An Egghead of Eulers or An Ark of Archimedes' are competitive
with any of these families. But one person does not
a family make.