RPI for Division I Women's Soccer - RPI: Formula (2024)

Updated March 2023

WHAT IS THE RPI?

The Rating Percentage Index is a mathematical system for rating sports teams. 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 expanded use of the RPI to other sports, with the following Division I sports now using it: men's and women's soccer, men's and women's volleyball, women's field hockey, men's and women's ice hockey, men's and women's lacrosse, baseball, softball, and women's water polo. Interestingly, the NCAA stopped using the RPI for men's basketball beginning with the 2018-19 season, replacing it with the much more complex NET system. In addition, it now uses that system for women's basketball. 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 some from sport to sport. The central structure 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, plus bonus and penalty adjustments. It considers only games against Division I opponents. Below, "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 penalty kicks 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) = (8 + 2)/20 = 10/20 = .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 (Against Other Teams)

Element 2 measures a team's opponents' average winning percentage (against teams other than Team A). The NCAA's stated purpose of Element 2, combined with Element 3, is to measure the strength of schedule against which Team A achieved its Element 1 Winning Percentage.

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 won the game, 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' Element 2s.

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. 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:

The table shows, for each year, 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 a result of using the 25%-50%-25% RPI formula. 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. Teams on average play about 17 games in a season, so for Element 2, the computation looks at about 17 teams' records and averages them out. 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 289 (17 x 17) 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 blue highlighted numbers show the average effective weights of the three elements covered by the table, when the three elements are incorporated into the RPI formula using the 25%-50%-25% ratios:

Element 1: 50.7 % -- roughly 50%

Element 2: 38.2% -- roughly 40%

Element 3: 11.0% -- roughly 10%

If you are having trouble understanding this, think of fruit salad. I want my fruit salad to consist of 50% cantaloupe (Element 1), 40% oranges (Element 2), and 10% kiwi fruit (Element 3). 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."

For those who are interested in how the average spreads and effective weights compare to what they were when there were overtimes:

With overtimes, the average spreads were:

Element 1: 0.9359

Element 2: 0.3633

Element 3: 0.2079

If you compare these spreads to the yellow highlighted spreads at the bottom of the above table, you can see that with no overtimes, the spreads are smaller, which means that the ratings are more compressed.

With overtimes, the effective weights were:

Element 1: 50.0%

Element 2: 38.8%

Element 3: 11.1%

These are very similar to the no overtime effective weights and likewise match the rounded off 50%-40%-10% effecctive weights

BONUS AND PENALTY ADJUSTMENTS

The formula described above produces Team A's unadjusted (or "basic" or "normal" or "original") RPI. Once the NCAA has calculated teams' unadjusted RPIs, 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. On this website, when I refer to the RPI, I am referring to the adjusted RPI, as that is how it most commonly is referred to.

The bonus and penalty structures and amounts vary from sport to sport. The committee for a particular sport sets the basic structure and amounts for that sport. In setting these, the committee, among other things, typically 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 unadjusted RPI ratings evolve over time, especially as additional schools sponsor a sport, the staff periodically, between seasons, can re-calibrate the bonus and penalty amounts to reflect changes needed in order for the bonus and penalty amounts to reflect the number of positions the committee decided it wants teams to rise or descend. The staff's re-calibration revisions ordinarily are in the range of an 0.0001 to 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.

Here are the Division I Women’s Soccer bonus and penalty adjustment structure and amounts that have been in effect since 2015. The bonuses and penalties are only for non-conference games. And, there are no bonuses or penalties for non-Division I games.

The Women's Soccer Committee has not disclosed publicly the number of positions it wants teams to rise or descend based on bonuses or penalties. 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 poor losses ... the penalties mirror the bonuses.

For 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 amounts for descents of 1 and 2 positions 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 URPI 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. Thus it appears (although I have seen no direct verification of it from the NCAA) that the NCAA staff understands an advance of 1 position to be the same as the average rating space between adjacent teams in the unadjusted RPI 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).

The above current bonus and penalty structure for soccer suggests that the Women’s Soccer Committee’s fundamental instruction to the NCAA staff about the the bonus and penalty amounts has been that the maximum bonus for a win -- an away win against a team ranked #1 through #40 -- should be equivalent to a 2 rank position improvement and for a tie -- an away tie against a team ranked #1 through #40 -- should be equivalent to a 1 rank position improvement. It further appears that the NCAA staff then fills in the remaining bonus and penalty amounts within the structure in a manner consistent with those instructions. Finally, the table suggests that the NCAA staff, prior to the 2015 season, identified 0.0013 as the rating difference equal to a 1 rank position difference.

The following table confirms that in the years leading up to 2015, a rating difference of 0.0013 was equal, on average, to a 1 ranking position change. This table is based on actual ratings from the years from 2007 to 2021, when teams were playing overtime games, since that is the system the current bonus and penalty adjustments are based on.

In this table, the "Average Difference" column shows, for each year, the average unadjusted RPI rating difference between teams, based on the end of regular season ratings. 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 the unconfirmed assumption that the NCAA staff would use the average differences over some number of years in determining a rating difference equal to a 1 ranking position change. The bottom two green rows show years in which the NCAA staff made changes to the bonus and penalty amounts. In 2010, there was a change from earlier amounts to a base amount of 0.0012. (There are not enough data available from earlier years to be able to tell if historic data support this amount.) In 2015, there was a change to a base amount of 0.0013. The 2015 base amount is consistent with what the above table shows as the appropriate rating difference to match a 1 rank position change.

Also, since there were bonus and penalty amount adjustments in 2010 and 2015, it is possible there is a 5-year review cycle. If so, then the next time for staff adjustments ordinarily would have been the 2020 season. Based on the table, however, 2020 adjustments would not have been appropriate, as the average rating gap between teams still was 0.0013. Consistent with this, there was no change in 2020.

As the upper most table on this page and the subsequent comparisons of the overtime to the no overtime data show, however, with the change to no overtimes, the ratings have become slightly more compressed. This suggests the possibility of a reduction in the bonus and penalty amounts in the future. The following table, based on RPI ratings since 2010 re-computed as though there had been no overtimes, suggests a reduction from the current 0.0013 to 0.0012:

How important are the bonuses and penalties?

The following two tables cover the years from 2007 through 2021 (excluding the 2020 Covid year) and use the actual with-overtime data from those years.

The first table, in the column on the left, shows the difference amounts between the end-of-regular-season unadjusted RPI rankings and adjusted RPI rankings for the Top 70 RPI teams. The Top 70 includes all teams that are potential competitors for at large positions in the NCAA Tournament (plus some additional teams that realistically are not competitors). The second column shows the numuber of cases in which the difference amount matched the amount in the Difference Amount column. The third column shows the number of cases as a percentage of all cases. And, the fourth column shows the average number of cases each year for the particular difference amount.

The second table shows aggregate numbers of cases and the corresponding percentages for 0 to 1, 0 to 2, 0 to 3, 0 to 4, and 0 to 5 rank position changes as a result of the bonus and penalty adjustments.

As the second table shows, almost two-thirds of teams experience changes of 1 or fewer rank positions and more than 90 percent have changes of 3 or fewer. Given the inherent limitations of rating systems, differences of these numbers of rank positions have little meaning.

How Has the Bonus and Penalty Structure Changed Over Time?

From 2007 to 2009 (and perhaps earlier), the bonus and penalty structure was as follows, apparently considering 0.0016 as a 1 rank position difference. The bonuses and penalties applied to both conference and non-conference games.

Win v RPI 1 to 40: 0.0032 (away), 0.0030 (neutral), 0.0028 (home)

Tie v RPI 1 to 40: 0.0016 (away), 0.0014 (neutral), 0.0012 (home)

Win v RPI 41 to 80: 0.0018 (away), 0.0016 (neutral), 0.0014 (home)

Tie v RPI 41 to 80: 0.0012 (away), 0.0010 (neutral), 0.0008 (home)

Tie v RPI 135 to 205: -0.0008 (away), -0.0010 (neutral), -0.0012 (home)

Loss v RPI 135 to 205: -0.0014 (away), -0.0016 (neutral), -0.0018 (home)

Tie v RPI 206 and poorer: -0.0012 (away), -0.0014 (neutral), -0.0016 (home)

Loss v RPI 206 and poorer: -0.0028 (away), -0.0030 (neutral), -0.0032 (home)

In 2010, there was a change to the following structure, apparently using 0.0012 as a 1 rank position difference. The bonuses and penalties applied to both conference and non-conference games.

Win v RPI 1 to 40: 0.0024 (away), 0.0022 (neutral), 0.0020 (home)

Tie v RPI 1 to 40: 0.0012 (away), 0.0010 (neutral), 0.0008 (home)

Win v RPI 41 to 80: 0.0018 (away), 0.0016 (neutral), 0.0014 (home)

Tie v RPI 41 to 80: 0.0006 (away), 0.0004 (neutral), 0.0002 (home)

Tie v RPI 135 to 205: -0.0002 (away), -0.004 (neutral), -0.0006 (home)

Loss v RPI 135 to 205: -0.0014 (away), -0.0016 (neutral), -0.0018 (home)

Tie v RPI 206 and poorer: -0.0008 (away), -0.0010 (neutral), -0.0012 (home)

Loss v RPI 206 and poorer: -0.0020 (away), -0.0022 (neutral), -0.0024 (home)

In 2012, the Committee made two changes in the structure, but not the amounts. It eliminated bonuses and penalties for conference games. And, it changed the penalty tiers to the teams ranked 41 to 80 positions from the bottom of the rankings and the bottom 40 teams in the rankings.

In 2015, the staff changed the amounts based on 0.0013 as a 1 rank position, resulting in the current structure and amounts set out in the table above on this page.