### RPI: Formula

Updated September 2018
WHAT IS THE RPI?

The Rating Percentage Index is a mathematical rating system based on the number of games teams have won, lost, and tied.  The NCAA began developing the RPI in the late 1970s for use in selecting teams to participate in the NCAA Division I Men's Basketball Championship.  The first actual use of the RPI for men's basketball was in 1982.  Over time, the NCAA has changed how it calculates the RPI.  The NCAA also has expanded its use to other sports, with the following Division I sports now using the RPI: men's and women's soccer, men's and women's volleyball, women's field hockey, men's ice hockey, men's and women's lacrosse, baseball, softball, and women's water polo.  Interestingly, the NCAA has stopped using the RPI for basketball as of the 2018-19 season, replacing it with a much more complex system.  It is not yet known whether the NCAA will make a comparable change for other sports at some point in the future.

The NCAA first used the RPI for Division I women's soccer in 1997.

The way in which the NCAA computes the RPI varies from sport to sport.  There even are variations in the RPI formula between men's and women's soccer.  The essence of the RPI, however, is the same for all sports.

This website deals only with the RPI as used for Division I women's soccer.

COMPUTING THE RPI

The RPI consists of three Elements, supplemented by an adjustment process.  It considers only games against Division I opponents.  In the following discussion, "Team A" refers to the team whose RPI is being computed.

Element 1:  Team's Winning Percentage.

Element 1 of the RPI compares the number of games Team A has won and tied to the total games Team A has played.  For purposes of this Element, the formula treats a tie as half a win and half a loss.  The formula for Element 1 is:

(W + 1/2T)/((W + 1/2T) + (L + 1/2T))

which simplifies to

(W + 1/2T)/(W + L + T)

In this formula, W is Team A's wins; T is Team A's ties; and L is Team A's losses.  Games determined by a penalty kick shootout are considered ties.

So, if Team A has a record of 8 wins, 8 losses, and 4 ties, Element 1 of its RPI is

(8 + (1/2 x 4))/(8 + 8 + 4) = .5000

Element 1 tells only Team A's wins and ties compared to its games played.  It tells nothing about the strength of Team A's opponents.  Thus, as an example, Element 1 for a team with an 8-8-4 record against the top 20 Division I teams will be .5000 and Element 1 for a team with an identical record against the bottom 20 Division I teams also will be .5000.

Element 2: Opponents' Average Winning Percentage.

Element 2 measures a team's opponents'  average winning percentage.  The purpose of Element 2, combined with Element 3, is to measure the strength of schedule against which Team A achieved its Element 1.

To determine Team A's opponents' average winning percentage, the NCAA first computes, for each of Team A's opponents, the opponent's wins and ties as compared to the opponent's total games played, in the same way it does the calculation for Team A's Element 1.  The only difference is that the NCAA excludes the opponent's games against Team A itself.  Thus this first part of the computation determines each opponent's Element 1 based on games played against teams other than Team A.  (For discussion about this method for computing Element 2, see the "RPI: Element 2 Issues" page.)

So, if Team A played an opponent once and defeated that opponent, the portion of Element 2 of Team A's RPI attributable to that opponent is determined by the following formula, in which O stands for "Opponent's":

(OW + 1/2OT)/(OW + (OL - 1) + OT)

Note that in the denominator, 1 is subtracted from the Opponent's losses.  This is because the Opponent lost to Team A, and by rule the result of the Opponent against Team A is not to be considered in determining that Opponent's contribution to Element 2 of Team A's RPI.

Likewise, if Team A tied the Opponent, the portion of Element 2 attributable to that Opponent is determined by the following formula:

(OW + 1/2(OT - 1))/(OW + OL + (OT - 1))

And, if Team A lost to the Opponent, the portion of Element 2 attributable to that Opponent is:

((OW - 1) + 1/2OT)/((OW - 1) + OL + OT)

Once the NCAA does this calculation for each opponent, it then computes the average of the numbers so computed for all of Team A's opponents.  This average is Element 2 of Team A's RPI.

Note that the NCAA does not simply add up the different opponents' wins, losses, and ties and then do a single calculation of wins and ties in relation to games played.  Rather, it does a calculation for each opponent and then averages the results.  The NCAA uses this averaging method to take into account the fact that different opponents play different numbers of games: By averaging, the NCAA assures that each opponent's contribution to Team A's Element 2 is weighted the same as each other opponent's contribution.

Also, if Team A plays multiple games against an opponent, then the opponent's winning percentage against teams other than Team A is counted multiple times in determining Team A's opponents' average winning percentage.

Element 3:  Team A's Opponents' Opponents' Average Winning Percentage.

Element 3 of the RPI measures a team's opponents' opponents' average winning percentage using, for each Team A opponent, the same method to determine that opponent's opponents' winning percentage as used in computing Team A's Element 2.  Thus another way to describe Team A's Element 3 is to say it is the average of Team A's opponents' Elements 2.

Calculation of RPI.

Once the NCAA has calculated each of these Elements, it combines them to determine the variously called "basic" or "normal" or "original" or "unadjusted" RPI.  I call it the Unadjusted RPI or URPI. The formula for determining the Unadjusted RPI is:

(Element 1 + (2 x Element 2) + Element 3)/4

At first glance, this looks like the RPI formula gives Team A's strength of schedule (Elements 2 and 3) three times the impact on the RPI that Team A's winning record (Element 1) has, since Element 1 counts for 25% of the formula's apparent weight, Element 2 counts for 50%,and Element 3 counts for 25%.  In effect, however, this is not true.  The following table shows why:

This table shows, for each year from 2007 through 2017, the high and low of each of the three RPI elements.  It then shows the difference (spread) between the high and low for each element.  And, based on the spread for each element, it shows the effective weight of each element as incorporated into the RPI formula's 25%-50%-25% apparent weights.  The bottom row of the table shows the averages for the element highs and lows, the element spreads, and the element effective weights.

As the table shows, the spreads for the three elements grow smaller when progressing from Element 1 to Element 3.  The reason for the diminishing spreads is obvious, if one thinks about it.  The computation of Element 1 looks at one team's record.  Individual teams' records reasonably can range from undefeated (an RPI Element 1 of 1.0000) to all losses (an RPI Element 1 of 0.0000), for a maximum reasonable (though not average) spread of 1.0000.  For Element 2, the computation looks at about 19 teams' records and averages them out.  Teams, on average, play about 19 games in a season.  With this many teams' records being used for Element 2, nearly all of the teams are going to have some wins and some losses, so the high Element 2 is going to be less than 1.0000 and the low is going to be higher than 0.0000.  Similarly, for Element 3 the computation looks at about 361 (19 x 19) teams' records.  This inclusion of a very large number of teams' records produces Element 3 numbers that are even less at the extremes than for Element 2, making Element 3's maximum reasonable (and average) spread smaller than for Element 2 and much smaller than for Element 1.

At the bottom right of the table, the yellow highlighted numbers show the average effective weights of the three elements over the 11 year period covered by the table, when the three elements are incorporated into the RPI formula using the 25%-50%-25% formula ratios:

Element 1:  49.6% -- roughly 50%

Element 2:  38.9% -- roughly 40%

Element 3:  11.5% -- roughly 10%

If you are having trouble understanding this, think of fruit salad.  I want my fruit salad to consist of 50% cantaloupe, 40% oranges, and 10% kiwi fruit.  To do that, I compare the fruit sizes and figure out that the right ratio of ingredients is 1 cantaloupe to 2 oranges to 1 kiwi fruit.  In this analogy, 1 canteloupe = 1 x RPI Element 1; 2 oranges = 2 x RPI Element 2; and 1 kiwi fruit = 1 x RPI Element 3.

The 50-40-10 percentages suggest that the NCAA adopted the 1:2:1 weights in the formula for the three Elements in order to have a team's winning percentage count for approximately half the team's RPI (Element 1's roughly 50% effective impact) and the team's strength of schedule count for the other half of the team's RPI (Element 2's roughly 40% effective impact plus Element 3's roughly 10% effective impact).  In a January 23, 2009 Memorandum from the NCAA's Associate Director of Statistics to the Division I Men's Basketball Committee, the NCAA confirmed that this is its intention:  "About half of the rating is based on winning percentage and the other half on strength of schedule.  Winning percentage (Factor I) only receives a 25 percent weighting although its real strength is larger.  There always is a far wider gap in the rankings between the top and bottom teams in this category than between the first and last in Factors II and III."

The formula described above produces Team A's Unadjusted (or "basic" or "normal" or "original") RPI.  Once the NCAA has calculated the RPI rating amounts, it then adjusts them by adding bonuses for "good" wins and ties and subtracting penalties for "poor" losses and ties, to produce the Adjusted RPI.  As between the URPI and the ARPI, the ARPI appears to be what the Committee uses in its decision-making process, although the Committee has access to all three RPI elements, to the URPI, and to the ARPI.  (The Committee also uses a variant of the RPI called the Non-Conference RPI and also has access to its three elements, to the UNCRPI, and to the ANCRPI.)

From time to time, the Women's Soccer Committee and, in certain limited situations, the NCAA staff change the bonus and penalty structure and amounts that lead to the Adjusted RPI. The Committee sets the basic structure for the bonus and penalty adjustments.  In setting the structure, the Committee typically (and perhaps always) tells the staff how many positions it wants a team to rise in the rankings for a good result or descend in the rankings for a poor result.  The staff then identifies the adjustment amount that will achieve the intended rise or descent.  Since the RPI ratings evolve over time, especially as additional schools sponsor Division I women's soccer, the staff periodically, between seasons, can re-calibrate the bonus and penalty amounts to reflect changes needed in the bonus and penalty amounts in order for them to reflect the number of positions the Committee has decided it wants teams to rise or descend.  The staff's re-calibration revisions ordinarily are in the range of an 0.0001 or 0.0002 change in bonus and/or penalty amounts.  These are the only changes the staff is allowed to make on its own.  The Committee must approve all other changes.  (I suspect, but don't know, that the staff checks the need for re-calibrations every five years.)

For Division I women's soccer, the Women's Soccer Committee has not disclosed the number of positions it wants teams to rise or descend based on bonuses or penalties.  It also has not disclosed the amounts of the bonus and penalty adjustments.  For Division I women's volleyball, however, the Women's Volleyball Committee has disclosed the following:

For wins over teams ranked 1-25 by the Unadjusted RPI, a team is to receive a rating bonus equivalent to an advance of 2 positions in the rankings;

For wins over teams ranked 26-50 by the Unadjusted RPI, a team is to receive a rating bonus equivalent to an advance of 1 position in the rankings;

For scheduling at least half of its non-conference games against teams in the top 75 of the Unadjusted RPI, a team is to receive a rating bonus equivalent to an advance of 2 positions in the rankings.

For poor losses and weak scheduling, the penalties mirror the bonuses.

B
y going to the NCAA's RPI Archive and reviewing the Team Rankings (all games) and the Team Sheets reports for women's volleyball, I have been able to determine that for Division I women's volleyball, the bonus amount for an advance of 1 position in the URPI ratings is 0.0013 and for an advance of 2 positions is 0.0026.  The penalty amount descents likewise are 0.0013 and 0.0026.  Further, by determining the difference between the ratings of the #1 ranked and the most poorly ranked team each year and then determining the average rating space between adjacent teams across the whole spectrum of the year's ratings, I have determined that 0.0013 is the average rating space between adjacent teams, which matches the 0.0013 amount for a change of 1 position.  Thus it appears (although I have seen no direct verification of it in NCAA documents) that the NCAA staff understands a rating bonus equivalent to an advance of 1 position to be the same as the average rating space between adjacent teams in the URPI rankings, with the average being taken across the whole spectrum of teams.  Thus if A is the rating of the #1 ranked team, B is the rating of the most poorly ranked team, and C is the number of teams sponsoring the sport, the formula for the rating adjustment equal to a 1 ranking position change is:  (A - B)/(C-1).

Applying this to women's soccer, the following table provides information on the rating adjustment equal, on average, to a 1 ranking position change:

In this table, the "Average Difference" column shows, for each year, the average URPI rating space between teams, based on the end of the regular season ratings (including conference tournaments).  The next columns show averages of these differences over the noted numbers of years.  I've shown the average differences over numbers of years based on an assumption that the NCAA staff would use the average differences over a number of years in setting a rating space equal to a 1 ranking position change.  I do not know, however, what number of years the staff would use.  The green rows show the years in which there have been changes to the bonus and penalty amounts or structure.  I discuss these changes in detail below.  In 2015, the staff made an adjustment in the bonus and penalty amounts with the base amount for one position change going from the 0.0012 set in 2010 to 0.0013.  This appears consistent with what the above table shows.

The following table shows the bonus and penalty amounts in effect starting probably in 2000 and definitely in effect from 2007 through 2009.  (It appears there were no bonus and penalty adjustments prior to 2000.)  I generally refer to this as the 2009 bonus and penalty structure:

 Result Location Opponents'  URPI Rank Type of Award Bonus/Penalty Scale Amount Win Away 1-40 Bonus (Highest point value awarded) .0032 Win Neutral 1-40 Bonus .0030 Win Home 1-40 Bonus .0028 Win Away 41-80 Bonus .0018 Win Neutral 41-80 Bonus .0016 Win Home 41-80 Bonus .0014 Tie Away 1-40 Bonus .0016 Tie Neutral 1-40 Bonus .0014 Tie Home 1-40 Bonus .0012 Tie Away 41-80 Bonus .0012 Tie Neutral 41-80 Bonus .0010 Tie Home 41-80 Bonus (Lowest point value awarded) .0008 Tie Away 135-205 Penalty (Lowest penalty imposed) -.0008 Tie Neutral 135-205 Penalty -.0010 Tie Home 135-205 Penalty -.0012 Tie Away 206-301 Penalty -.0012 Tie Neutral 206-301 Penalty -.0014 Tie Home 206-301 Penalty -.0016 Loss Away 135-205 Penalty -.0014 Loss Neutral 135-205 Penalty -.0016 Loss Home 135-205 Penalty -.0018 Loss Away 206 and poorer Penalty -.0028 Loss Neutral 206 and poorer Penalty -.0030 Loss Home 206 and poorer Penalty (Greatest penalty imposed) -.0032

Ideally, I would have access to the end-of-regular-season ratings for years prior to 2007, which is the year in which my data base begins.  The NCAA's RPI Archive, however, does not have those ratings.  It does, however, have "Final" ratings for years prior to 2007, which include NCAA Tournament games in their data base.  For the Final ratings, the average rating difference equivalent to a 1 ranking position change is as as follows: 2000 = 0.0016; 2001 = 0.0016; 2002 = 0.0015; 2003 = 0.0019; 2004 = 0.0017; 2005 = 0.0015; 2006 = 0.0014.  Based on these numbers, it seems reasonable to believe that the NCAA staff developed the above table using a rating difference of 0.0016 as equal to 1 ranking position.  Using that amount, the bonus and penalty structure would have been consistent with the following:

For an away win over a team ranked 1-40 by the URPI, a team should receive a rating bonus equivalent to 2 ranking positions

For an away win over a team ranked 41-80 by the URPI, a team should receive a rating bonus equivalent to 1 1/8 ranking positions

For an away tie over a team ranked 1-40 by the URPI, a team should receive a rating bonus equivalent to 1 ranking position

For an away tie over a team ranked 41-80 by the URPI, a team should receive a rating bonus equivalent to 3/4 ranking positions

For the above tiers but neutral and home sites, the instruction appears to have been to reduce the corresponding rating bonus by the equivalent of 1/8 ranking positions (0.0002) for neutral site games and 1/4 ranking positions (0.0004) for home games.

For poor ties and losses, the penalty amounts structure mirrors the bonus structure, except that the penalties apply to tiers of teams as shown in the table.

The following table shows the bonus and penalty amounts in effect for the 2010 and 2011 seasons.  I generally refer to this as the 2010 bonus and penalty structure:

 Result Location Opponents'  URPI Rank Type of Award Bonus/Penalty Scale Amount Win Away 1-40 Bonus (Greatest point value awarded) .0024 Win Neutral 1-40 Bonus .0022 Win Home 1-40 Bonus .0020 Win Away 41-80 Bonus .0018 Win Neutral 41-80 Bonus .0016 Win Home 41-80 Bonus .0014 Tie Away 1-40 Bonus .0012 Tie Neutral 1-40 Bonus .0010 Tie Home 1-40 Bonus .0008 Tie Away 41-80 Bonus .0006 Tie Neutral 41-80 Bonus .0004 Tie Home 41-80 Bonus (Lowest point value awarded) .0002 Tie Away 135-205 Penalty (Lowest penalty imposed) -.0002 Tie Neutral 135-205 Penalty -.0004 Tie Home 135-205 Penalty -.0006 Tie Away 206-301 Penalty -.0008 Tie Neutral 206-301 Penalty -.0010 Tie Home 206-301 Penalty -.0012 Loss Away 135-205 Penalty -.0014 Loss Neutral 135-205 Penalty -.0016 Loss Home 135-205 Penalty -.0018 Loss Away 206 and poorer Penalty -.0020 Loss Neutral 206 and poorer Penalty -.0022 Loss Home 206 and poorer Penalty (Greatest penalty imposed) -.0024

Comparing this table to the "Average Differences" table above, it appears to be based on the NCAA staff having determined that, at that time, a rating difference of 0.0012 was equivalent to a change of one ranking position.  Thus the basic structure became:

For an away win over a team ranked 1-40 by the URPI, a team should receive a rating bonus equivalent to 2 ranking positions

For an away win over a team ranked 41-80 by the URPI, a team should receive a rating bonus equivalent to 1.5 ranking positions

For an away tie over a team ranked 1-40 by the URPI, a team should receive a rating bonus equivalent to 1 ranking position

For an away tie over a team ranked 41-80 by the URPI, a team should receive a rating bonus equivalent to 0.5 ranking positions

For the above tiers but neutral and home sites, the instruction appears to have been to reduce the corresponding rating bonus by the equivalent of 1/6 ranking positions (0.0002) for neutral site games and 1/3 ranking positions (0.0004) for home games.

For poor ties and losses, the penalty amounts structure mirrors the bonus structure, except that the penalties apply to tiers of teams as shown in the table.

The following table shows the bonus and penalty amounts in effect for the 2012 through 2014 seasons.  The amounts are the same as for the 2010 bonus and penalty structure, and so appear to be based on 0.0012 being equivalent to a 1 ranking position change.  Unlike in earlier seasons, however, beginning in 2012 the bonuses and penalties are only for results in non-conference games.  In addition, it's important to note that URPI Rank "tiers" for the penalties changed to tiers of the bottom 40 teams (higher penalties) and the next to bottom 40 teams (lower penalties), which mirrors the bonus tiers.  I generally refer to this as the 2012 bonus and penalty structure:

 Result Location Opponents'  URPI Rank Type of Award Bonus/Penalty Scale Amount Win Away 1-40 Bonus (Greatest point value awarded) .0024 Win Neutral 1-40 Bonus .0022 Win Home 1-40 Bonus .0020 Win Away 41-80 Bonus .0018 Win Neutral 41-80 Bonus .0016 Win Home 41-80 Bonus .0014 Tie Away 1-40 Bonus .0012 Tie Neutral 1-40 Bonus .0010 Tie Home 1-40 Bonus .0008 Tie Away 41-80 Bonus .0006 Tie Neutral 41-80 Bonus .0004 Tie Home 41-80 Bonus (Lowest point value awarded) .0002 Tie Away 244-283 Penalty (Lowest penalty imposed) -.0002 Tie Neutral 244-283 Penalty -.0004 Tie Home 244-283 Penalty -.0006 Tie Away 284 and poorer Penalty -.0008 Tie Neutral 284 and poorer Penalty -.0010 Tie Home 284 and poorer Penalty -.0012 Loss Away 244-283 Penalty -.0014 Loss Neutral 244-283 Penalty -.0016 Loss Home 244-283 Penalty -.0018 Loss Away 284 and poorer Penalty -.0020 Loss Neutral 284 and poorer Penalty -.0022 Loss Home 284 and poorer Penalty (Greatest penalty imposed) -.0024

The following table shows the bonus and penalty amounts in effect since the 2015 season.  These are only slightly different than the amounts for the 2012 and 2013 seasons, and appear to be based on the NCAA staff having identified a change in the rating difference equal to a 1 ranking position change, with the change being from 0.0012 to 0.0013.  Thus the basic pattern of 2 - 1.5 - 1 - 0.5 ranking position changes appears to be the same as for the 2010 bonus and penalty structure.  And again, the bonuses and penalties apply only to non-conference games.

And there's one last item: The RPI formula, from 2007 through the 2014, also imposed penalties for ties and losses against non-Division I teams as though those teams were ranked in the tier susceptible to the higher penalties.  Beginning with the 2015 season, however, the programming for the RPI formula changed so that it no longer imposed these penalties.  This eventually was reflected  In the Division I Women's Soccer 2016 Pre-Championship Manual, Appendix C, which describes the bonus and penalty system.  There, the Committee states, for the first time: "There is no penalty for losing to a non-DI team."

How important are the bonuses and penalties?  The following table, using the 2015 season and 2015 ARPI formula, shows the difference between the Unadjusted RPI rankings and the Adjusted RPI rankings, for the Top 60 Unadjusted and Adjusted RPI teams.  The top 60 includes all teams, plus a few, that might be under consideration for at large selections and seeding for the NCAA Tournament and thus is a key group to consider for RPI evaluation purposes:

The following table summarizes these data, showing that the 2015 adjustments make little difference, as compared to the unadjusted RPI.  About a third of the teams experienced no rank change from the URPI; about two thirds experienced a change of at most one rank position; and more than 85% experienced a change of at most two rank positions.

Thus the bonus and penalty adjustments, given the the current bonus and penalty amounts and structure, are of relatively little significance from an RPI perspective.

EDITORIAL COMMENT

The Women's Soccer Committee and NCAA staff ordinarily have not made public announcements in advance when there has been a change in the RPI formula, such as a change to the bonus/penalty system.  Neither has the Committee, during the 11 years I have been studying the RPI, explained to the public why it has made changes.  Nor has the NCAA staff make a public announcement in advance when it re-calibrated the bonus and penalty amounts.  Why the schools put up with this is a mystery to me, since it effectively means that the schools do not know the rules that will govern at large selections and that will affect seeding for the NCAA Tournament until part way through the season, if at all.  Hopefully, the NCAA will change this practice and will announce formula changes and the reasons for them in advance of the first season they will affect.  The staff is very busy, but this does not seem too much to ask.