# Statistics Glossary

This is an ongoing project.

Base Runs (BsR):  Times on Base * Score Rate + Home Runs

Designed by sabermetrician David Smyth, Base Runs is an incredibly flexible and highly accurate dynamic run estimator that does a phenomenal job at modeling the run-scoring process.  BsR is expressed as A*(B/(B+C))+D, where “A” are the times on base, “B” is the advancement factor, “C” are the outs made, and “D” are home runs.  The “B” factor can be tweaked in order to maximize accuracy, and B/(B+C) is the estimated score rate.  The most basic BsR formula is as follows:

A: H + BB – HR

B: (1.4*TB – .6*H – 3*HR + .1*BB) * X

C: AB – H

D: HR

Where “X” is the multiplier that calibrates the “B” factor to match the league runs scored.  “X” is equal to (Runs – HR)*C/(A – Runs + D).  The “X” multiplier for MLB 2005-2009, for example, is 0.998.  BsR can also be used to find the marginal run values of each event, also known as Linear Weight values.  More rigorous equations (such as Tom Tango’s) that include other events will increase accuracy as well.  BsR works fine for pitchers, but not for hitters- this is because the player’s advancement and out factors interact with one another.  This can be fixed by using a theoretical team approach.  The best article I am aware of on BsR is by Brandon Heipp, which you can find here.

Batting Average (BA, AVG): H / AB

The most recognizable rate statistic used in mainstream analysis, batting average is a descriptive statistic and not a value one.  In other words, it is inadvisable to use it in regards to player value because it fails to account for the impact of each hit type.

Batting Average on Balls In Play (BABIP): (H – HR) / (AB – HR – K + SF)

This tells us the rate at which a player’s balls in play land for a hit.  The league average typically hovers around .290-.300 for both hitters and pitchers, and anything above or below that is generally attributed to good or bad “luck.”  Vöros McCracken found that pitchers have little influence on their BABIP, suggesting that Earned Run Average (ERA) and Walks and Hits per Innings Pitched (WHIP) are heavily reliant on the pitcher’s defensive support.  This is not particularly true, however, as pitchers do have control over their batted ball types, which will in turn influence their BABIP (ground balls and fly balls fall for hits at different rates).  Pitchers seem to have considerably less control over their BABIP than hitters do.

Defensive Efficiency Ratio (DER): (BFP – H  – BB – HBP – K – ROE) / (BFP – HR – K – BB – HBP)

DER attempts to tell us how efficient a defense is at turning balls in play into outs.  It can be improved upon by accounting for batted ball types, due to the fact that not all teams face the same ground ball/fly ball ratio.

Isolated Power (ISOP): SLG – AVG

ISO is meant to give us an idea of how much power a player has by subtracting the player’s slugging percentage- his rate of hitting for extra bases- by his batting average, his overall rate of hits.  In its simplest form, ISO can be expressed as (2B + 2*3B + 3*HR) / AB.  Major League average generally sits around .150.

Linear Weights (LWTS): rv*1B + rv*2B + rv*3B + rv*HR + rv*ROE + rv*NIBB + rv*HBP – rv*Non-K Out – rv*K

Where “rv” stands for “run value.”  Popularized by Pete Palmer and John Thorn in The Hidden Game of Baseball, Linear Weights are the best way to evaluate an individual hitter’s performance.  Linear weight values are usually derived through run expectancy charts to determine the empirical marginal value of each event, although other approaches, such as a Markov Chain, are also used.  The values shown on this site are usually (if not always) derived through a Base Runs model using the “plus-one” method as outlined by Brandon Heipp.  LWTS are often expressed as runs above or below the league average, although they can be expressed as absolute runs (like BsR) as well.  A linear weights equation for MLB from 2005-2009, for example, looks like this:

LWTS = .49*1B + .78*2B + 1.07*3B + 1.41*HR + .50*ROE + .32*NIBB + .35*HBP – .29*(AB – H – ROE – K + SF) – .30*K

On-Base Percentage (OBP): (H + BB + HBP) / PA

On-Base Percentage, much like batting average, is a descriptive statistic.  It does exactly what it’s meant to do: tell us the rate at which a player (mostly) reaches base.  Just like batting average, it is inadvisable to use it in regards to player value because it too fails to account for hit types.  But it is a clear upgrade over batting average in some regards because of the inclusion of the walk.

Slugging Percentage (SLG): (1B + 2*2B + 3*3B + 4*HR) / AB

Slugging percentage is a clear improvement over batting average, in that it accounts for the impact of the extra base hit.  It does not, however, weight each event properly.  Four singles do not equal one home run, nor do two singles equal one double.  Just like batting average and on-base percentage, slugging percentage is a descriptive statistic.

On-Base Plus Slugging Percentage (OPS): OBP + SLG

If we have two descriptive measures in on-base percentage, which measures the player’s ability to get on base, and slugging percentage, which measures the player’s propensity to hit for extra bases, might adding the two together give us a measure of a player’s value?  In some ways, yes.  OPS works rather well at approximating the run-scoring process, but ultimately fails at assessing individual players due to the nature of the rates being added together.  On-base percentage uses plate appearances as its denominator, while slugging percentage uses at-bats.  Walks, therefore, will be undervalued.  Not only that, but because of the awkward weighting of slugging percentage, extra base hits are weighted improperly as well.  OPS is advisable for a quick-n’-dirty assessment of a player’s performance, but should never be used when something like wOBA is available.

Weighted On-Base Average (wOBA): (rv*1B + rv*2B + rv*3B + rv*HR + rv*ROE + rv*NIBB + rv*HBP) / (PA – IBB)

wOBA was designed by sabermetrician Tom Tango to help fix the weighting issues with OPS.  wOBA is, simply put, linear weights expressed as a rate statistic.  Because the value of the out in the numerator of on-base percentage is zero, the value of the out is absorbed into the numerator of wOBA.  The coefficients are then multiplied by a scale in order to make it match the league on-base percentage.  To understand the full gory technical details (and more), I suggest you read this primer.  wOBA is both a value and descriptive statistic, and is strongly recommended for player evaluation.

The wOBA formula for MLB 2005-2009, for example, would be as follows (set so the league average is exactly .330):

wOBA = (.89*1B + 1.23*2B + 1.56*3B + 1.95*HR + .91*ROE + .70*NIBB + .73*HBP) / (PA – IBB)

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