Formulas
 

Total Player Rating  - (Palmer)

TPR is actually the sum of several other metrics: adjusted batting runs, fielding runs and base stealing runs, all divided by the Runs Per Win Factor for that year, which is the average number of extra runs needed to generate a win over what the average player produces.  Generally that number is around 10, but ranges from 9-11.

The equations are:

Adjusted Batting Runs = ((.47)1B + (.78)2B + (1.09)3B + (1.40)HR + (.33)(BB + HBP) - (.25)(AB - H) - (.50)Outs on base), then adjusted for fielding position

Fielding Runs = (.20(Put outs + 2* assists - errors + DP)player's total - league average for the position * (team put outs - team Ks) * (innings played for the player/innings played for the team)

Base Stealing Runs = (.30)steals - (.60)caught stealing
 

Game Scores  - (James)

Game score is a metric devised by Bill James to show how dominating a pitcher was in any particular game.  To determine a starting pitcher's game score: (1) Start with 50 points.  (2) Add 1 point for each out recorded, so 3 points for every complete inning pitched. (3) Add 2 points for each inning completed after the 4th.  (4) Add 1 point for each strikeout.  (5) Subtract 2 points for each hit allowed.  (6) Subtract 4 points for each earned run allowed.  (7) Subtract 2 points for each unearned run allowed.  (8) Subtract 1 point for each walk.

The top game score in the history of baseball was Kerry Wood's one-hit, no walk, 20 strikeout performance against the Astros on May 6, 1998.  His game score was 105.
 

Runs Created  - (James)

There are 3 component equations to Runs Created.  Each is weighted by how much each event correlates with runs being scored.  Additionally, each event (walk, hit, steal, etc.) is weighted within each equation based on how much it correlates with scoring runs.  Here are the component equations:

A = Hits + Walks + Hit by Pitches - Caught Stealing - grounded into double plays

B = ((Walks - Intentional Walks + Hit by Pitches) * .24) + (Stolen Bases * .62) + ((Sacrifice Hits + Sacrifice Flies) * .5) + Total Bases - (Strikeouts * .03)

C = At Bats + Walks + Hit by Pitches + Sacrifice Hits + Sacrifice Flies

And here is the final equation for Runs Created:

((((C * 2.4) + A) * ((C * 3) + B))/(C * 9)) - (C * .9)

Additional adjustments are made when statistics with men in scoring position are available: divide a player's (or team's) total home runs by the number of at bats.  Multiply that number by the number of at bats with men on base to find the expected home runs in that situation.  Subtract the expected total from the real total and add that result to the raw runs created result.

Then multiply a player's (or team's) batting average by the number of at bats with runners on second or third (RISP) to determine the expected hits in that situation.  Subtract the expected number from the actual number and again add the result to the raw runs created total.

Round the final result to the nearest integer.  (whew!)
 
 

Total Average  - (Boswell)

Total Average is a metric devised by sportswriter Tom Boswell , primarily for a yearly evaluation of regular major leaguers by Inside Sport magazine.  It's one of the easier to calculate attempts at incorporating baserunning into an overall picture of a player's offensive contributions:

(((1B + (2Bx2) + (3Bx3) + (HRx4)) + HP + BB + SB) ­ CS) / ((AB - H) + CS + DP).
 
 
 

Pythagorean Theorem  - (James)

As anyone might surmise, there is a strong correlation between runs scored/runs allowed and wins/losses.  This equation expresses that correlation reasonably accurately.  Rarely do teams exceed or subordinate the theorem by more than 3 wins.  When they do, it's usually do to either luck or the bullpen, which in baseball terms, are sometimes synonymous.  Anyway, the equation is:

(runs scored * runs scored)/((runs scored * runs scored) + (runs allowed * runs allowed))

The result will give you the team's winning percentage, or at least what it should be based on how many runs they've scored and allowed.
 
 
 

Estimated Runs Produced (ERP)...  - (Johnson)

Paul Johnson came up with ERP in an effort to take into account and properly weigh every event a batter is involved in and express it in one neat number.  The formula correlates very strongly with Bill James Runs Created formulas, but is considerably less cumbersome:

ERP = (2 x (Total Bases + Walks + Hit by Pitches) + Hits + Steals - (.605 x (At Bats + Caught Stealing + GIDP - Hits))) x .16

Just so you don't think the constants were simply pulled out of a hat, they are the result of factoring the chances that each particular event (a walk, a double, etc.) will result in a run scored.