# Net run rate

Net Run Rate (NRR) is a statistical method used in analyzing team work and/or performance in the sport of cricket. It is the most commonly used method of ranking teams with equal points in limited overs league competitions, analogous to goal difference in association football.

The net run rate in a single game is the average runs per over that a team scores, minus the average runs per over that is scored against them. The net run rate in a tournament is the average runs per over that a team scores across the whole tournament, minus the average runs per over that is scored against them across the whole tournament.[1][2] Note this is not usually the same as the average of the net run rates from the individual matches in the tournament.

## Step by step explanation

A team's run rate (RR), or runs per over (RPO), is the average number of runs scored per over by the whole team in the whole innings (or the whole innings so far), ie. $\mbox{run rate }=\frac{\mbox{total runs scored}}{\mbox{total overs faced}}$.

So if a team scores 250 runs off 50 overs then their RR is $\frac{250}{50} = 5$. Note that as an over is made up of six balls, each ball is 1/6 of an over, despite being normally written in cricket's notation as .1 of an over. So if they got that same score off 47.5 overs, their RR would be $\frac{250}{47\frac{5}{6}} \approx 5.226$.

The concept of net run rate involves taking the opponents' final run rate away from the team's run rate, ie. $\mbox{net run rate }=\frac{\mbox{total runs scored}}{\mbox{total overs faced}}-\frac{\mbox{total runs conceded }}{\mbox{total overs bowled}}$.

Usually, runs and overs bowled are summed together throughout a season to compare teams in a league table. A team's overall NRR for a tournament is not defined as the sum or average of the NRR's from the individual matches, but as:

$\mbox{tournament net run rate }=\frac{\mbox{total runs scored in all matches}}{\mbox{total overs faced in all matches}} - \frac{\mbox{total runs conceded in all matches}}{\mbox{total overs bowled in all matches}}$

The exceptions to this are:

• if a team is bowled out, it is not the overs faced which their score is divided by; instead the full quota of overs to which it would have been entitled is used (e.g. 50 overs for an uninterrupted One Day International, and 20 overs for a Twenty20 match).
• if a match is interrupted and Duckworth-Lewis revised targets are set, the actual runs scored and overs faced are used for Team 2's innings (the side which batted second), but the revised targets and revised overs are used for Team 1's innings, ie.
• if the match is concluded, Team 1 is credited with 1 run less than the final Target Score for Team 2, off the total number of overs allocated to Team 2 to reach the target.[1]
• if the match is abandoned, but a result achieved, Team 1 is credited with Team 2's Par Score off the same number of overs faced by Team 2.[1]
• if a match is abandoned as a No Result, none of the runs scored or overs bowled count towards this calculation.

## Scenarios

All scenarios assume One Day International rules with 50 overs per side.

### 1. Side that bats first wins

• Team A bat first and score 287-6 off their full quota of 50 overs. Team B fail in their run chase, early losses causing them to struggle to 243-8 in their 50 overs.
• Team A's Run Per Over is $\frac{287}{50} = 5.74$
• Team B's Run Per Over is $\frac{243}{50} = 4.86$
• Team A's NRR for this game is 5.74 − 4.86 = 0.88. If this was the first game of the season, their NRR for the league table would be +0.88.
• Team B's NRR for this game is 4.86 − 5.74 = −0.88. If this was the first game of the season, their NRR for the league table would be −0.88.

### 2. Side that bats second wins

• Team A bat first and score 265-8 off their full quota of 50 overs. Team B successfully chase, getting their winning runs with a four with sixteen balls (2.4 of the 50 overs) remaining, leaving them on 267-5.
• Team A's rpo is $\frac{265}{50} = 5.30$
• Team B faced 47.2 overs, so their rpo is $\frac{267}{47\frac{2}{6}} \approx 5.64$
• Assuming that Team A and Team B had previously played as in the game in scenario one, the new NRR for Team A would be $\frac{287+265}{50+ 50}-\frac{243+267}{50+47.33} = \frac{552}{100}-\frac{510}{97.33} \approx 0.28$

### 3. Side that bats first is bowled out, side batting second wins

• Team A bat first and are skittled out for 127 off 25.4 overs. Team B reach the target off 30.5 overs, ending with 128/4.
• Despite Team A's runrate for the balls they faced being 127 / 25.667 = 4.95, because they were bowled out the entire 50 overs are added to their total overs faced tally for the tournament, and Team B are credited with having bowled 50 overs.
• Team B actually scored at a slower pace (128/30.833 = 4.15), however they managed to protect their wickets and win. Thus, only the 30.833 overs are added to the seasonal tally.
• Team A's NRR for this game is (127/50) − (128/30.833) = −1.61.
• Team B's NRR for this game is (128/30.833) − (127/50) = +1.61.
• If 25.667 had been used for Team A's overs total rather than 50, Team A would have finished the match with a positive match NRR, and improved tournament NRR, despite losing. (Similarly Team B with a worsened NRR, despite winning.)

### 4. Side that bats second is bowled out, side batting first therefore wins

• Team A bat first and set a formidable 295/5 off their complement of 50 overs. Team B never get close, being bowled out for 116 off 35.4 overs.
• 295 runs and 50 overs are added to Team A's tally.
• However, Team B, despite facing only 35.4 overs, have faced 50 overs according to the NRR calculations, and Team A have bowled 50 overs.

### 5. Both sides are bowled out, side batting first therefore wins

• Team A bat first, and manage 117 off 24 overs on a difficult playing surface. Team B fall agonizingly short, reaching 112 off 23.3 overs.
• In this case, both teams get 50 overs both faced and bowled in the overs column for the season, just as in example 1.

### 6. The game ends in a tie

• Runs and overs are added as in the examples above, with teams bowled out being credited with their full quota of overs. Thus, the match NRR will always be zero for both teams.

### 7. Interrupted games with revised targets

• In matches where Duckworth-Lewis revised targets are set due to interruptions which reduce the number of overs bowled, those revised targets and revised overs are used to calculate the net run rate for both teams.
• For example, in a 50-over World Cup first-round group match, Team A are dismissed for 165 in 33.5 overs.
• Team B progresses to 120-0, but play is halted after 18 overs due to rain.
• Six overs are lost, and the target is reset to 150, which Team B reach comfortably after 26.2 overs with only 2 wickets lost.
• Because the target was revised, 6 overs were lost and Team A were bowled out, Team A's total is reset to 149 from 44 overs, thus their RR $= \frac{149}{44} \approx 3.39$. Team B's RR, however, is computed as normal: $\frac{150}{26.33} \approx 5.70$.
• Computing the match NRR for Team A gives us 3.39 - 5.70 = -2.31. Team B's NRR is: 5.70 - 3.39 = 2.31.

### 8. Abandoned games recorded as No-Result

• Abandoned games are not considered, whatever the stage of the game at stoppage may be, and the scores in such games are immaterial to NRR calculations.

## Net Run Rate within a tournament

### Basic example

Most of the time, in limited overs cricket tournaments, there are round-robin groups among several teams, where each team plays all of the others. Just as explained in the scenarios above, the NRR is not the average of the NRRs of all the matches played, it is calculated considering the overall rate at which runs are scored for and against, within the whole group.

Let's take as an example South Africa's net run rate in the 1999 World Cup.

FOR

South Africa scored:

• Against India, 254 runs (for 6 wkts) from 47.2 overs.
• Against Sri Lanka, 199 runs (for 9 wkts) from 50 overs.
• Against England, 225 runs (for 7 wkts) from 50 overs.
• Against Kenya, 153 runs (for 3 wkts) from 41 overs.
• Against Zimbabwe, 185 runs (all out) from 47.2 overs.

In the case of Zimbabwe, because South Africa were all out before their allotted 50 overs expired, the run rate is calculated as if they had scored their runs over the full 50 overs. Therefore, across the five games, South Africa scored 1016 runs in a total of 238 overs and 2 balls (ie. 238.333 overs), an average run rate of 1016/238.333 = 4.263.

AGAINST

Teams opposing South Africa scored:

• India, 253 (for 5 wkts) from 50 overs.
• Sri Lanka, 110 (all out) from 35.2 overs.
• England, 103 (all out) from 41 overs.
• Kenya, 152 (all out) from 44.3 overs.
• Zimbabwe, 233 (for 6 wkts) from 50 overs.

Again, with Sri Lanka, England and Kenya counting as the full 50 overs as they were all out, the run rate scored against South Africa across the five games is calculated on the basis of 851 runs in a total of 250 overs, an average run rate of 851/250 = 3.404.

NET RUN RATE

South Africa's tournament net run rate is therefore 4.263 − 3.404 = +0.859.

### Change in NRR through a tournament

In the above example of South Africa at the 1999 World Cup, after their first match their tournament NRR was $\frac{\mbox{254}}{\mbox{47.333}} - \frac{\mbox{253}}{\mbox{50}}.$

After their second match their tournament NRR was $\frac{\mbox{254 + 199}}{\mbox{47.333 + 50}} - \frac{\mbox{253 + 110}}{\mbox{50 + 50}},$ which is the same as $\frac{\mbox{254}}{\mbox{97.333}} + \frac{\mbox{199}}{\mbox{97.333}} - \frac{\mbox{253}}{\mbox{100}} - \frac{\mbox{110}}{\mbox{100}}.$

After their third match it was $\frac{\mbox{254 + 199 + 225}}{\mbox{47.333 + 50 + 50}} - \frac{\mbox{253 + 110 + 103}}{\mbox{50 + 50 + 50}}.$ which is the same as $\frac{\mbox{254}}{\mbox{147.333}} + \frac{\mbox{199}}{\mbox{147.333}} + \frac{\mbox{225}}{\mbox{147.333}} - \frac{\mbox{253}}{\mbox{150}} - \frac{\mbox{110}}{\mbox{150}} - \frac{\mbox{103}}{\mbox{150}}.$

This shows that NRR can be calculated in two different ways:

• Sum all runs scored in the tournament, and divide this by the total number of overs faced in the tournament. Add together all runs conceded in the tournament, and divide by total number of overs bowled. Subtract bowled rate from batted rate. (The left hand side above.)
• Divide the runs scored in each innings by the total number of overs faced in the tournament, and the runs conceded in each innings by the total number of innings bowled in the tournament. Then add all batting rates and subtract all bowling rates. (The right hand side above.)

While less intuitive, the second way of thinking about this shows how the different innings' contributions to NRR compare with each other and change as the tournament progresses.

The different batting totals are all divided by the same figure, the total number of overs batted. However, as this increases with every match, so the contribution of each batted innings to tournament NRR reduces with every match. For example, the contribution to tournament NRR of the innings batted in the first match was 254/47.333 = 5.37 after the first match, 254/97.333 = 2.61 after the second match, and 254/147.333 = 1.72 after the third match.

This is similarly true for runs conceded from bowled overs, though the total numbers of overs bowled and batted are different at each stage, so the denominator for the bowled overs is different to the denominator for the batted overs. As the total number of overs batted is slightly less than the total number of overs bowled, each run scored contributes slightly more to tournament NRR than each run conceded.

The reducing contribution of each innings to tournament NRR is a reflection of the fact that tournament NRR can also be thought of as the weighted average of the run rates, and each time another match is played, the weights of the previous innings reduce, and so the influences of the previous innings on overall NRR reduce:

### Tournament NRR as weighted average of the run rates

As Run Rate = Runs scored/Overs faced, the runs scored by and against South Africa in each innings can be replaced in the formulas for NRR by Run Rate x Overs faced. For example, in the first match South Africa scored 254 runs from 47 overs and 2 balls, a rate of 5.37 runs per over. Therefore the total of 254 runs can be replaced by 5.37 runs per over x 47.333 overs. This gives a third way of finding tournament NRR:

After their second match their tournament NRR was $\left(5.37\times\frac{\mbox{47.333}}{\mbox{97.333}}\right) + \left(3.98\times\frac{\mbox{50}}{\mbox{97.333}}\right) - \left(5.06\times\frac{\mbox{50}}{\mbox{100}}\right) - \left(2.20\times\frac{\mbox{50}}{\mbox{100}}\right)$

                                   $=\left(5.37\times 48.6%\right)+\left(3.98\times 51.4%\right)-\left(5.06\times 50%\right)-\left(2.20\times 50%\right).$


After their third match it was $\left(5.37\times\frac{\mbox{47.333}}{\mbox{147.333}}\right) + \left(3.98\times\frac{\mbox{50}}{\mbox{147.333}}\right) + \left(4.50\times\frac{\mbox{50}}{\mbox{147.333}}\right) - \left(5.06\times\frac{\mbox{50}}{\mbox{150}}\right) - \left(2.20\times\frac{\mbox{50}}{\mbox{150}}\right) - \left(2.06\times\frac{\mbox{50}}{\mbox{150}}\right)$

                   $=\left(5.37\times 32.1%\right) + \left(3.98\times 33.9%\right) + \left(4.50\times 33.9%\right) - \left(5.06\times 33.3%\right) - \left(2.20\times 33.3%\right) - \left(2.06\times 33.3%\right).$


Therefore tournament NRR can alternatively be thought of as the weighted average of the run rates scored in each match (weighted by the lengths of the innings batted compared to the other innings batted), minus the weighted average of the run rates conceded in each match (weighted by the lengths of the innings bowled compared to the other innings bowled).

Each time another match is played, the weights of the previous innings reduce, and so the contributions of the previous innings to overall NRR reduce.

## Criticisms

### Tournament NRR calculation

A team's batted and bowled overs in a match count differently to tournament NRR

All overs batted in a tournament are given equal weighting when finding tournament NRR, and all overs bowled in a tournament are also given equal weighting. However, when the total number of overs batted is different to the total number of overs bowled, the weight for each over batted is different to the weight for each over bowled. This means that batted overs and bowled overs in the same match count differently towards tournament NRR.

For example, in the 2009 World Twenty20 Group D, as New Zealand had batted 6 overs and bowled 7 overs against Scotland, the runs they scored in each of the 20 overs batted against South Africa contributed 1/26th to their tournament NRR, while the runs conceded in each of the 20 overs bowled against South Africa contributed only 1/27th. In fact, the effect of the higher weight for the batting overs was so strong that despite scoring fewer runs than South Africa from the same number of overs, and hence having a negative match NRR and losing the match, the net contribution of this match to New Zealand's tournament NRR was actually positive (127/26 − 128/27 is positive).

Each over in a match counts differently for the two teams

Moreover, if two teams in a tournament have different total numbers of overs batted or bowled, then each innings in the match(es) between them will contribute differently towards their tournament NRRs. For example, in the 2009 World Twenty20 Group D, South Africa batted for 40 overs in total in their two matches, so their score of 128 from 20 overs against New Zealand contributed 128/40 = 3.20 to their tournament NRR, whereas New Zealand bowled for 27 overs in total in their two matches, so South Africa's score of 128 from 20 overs against them contributed −128/27 = −4.74 to New Zealand's tournament NRR.

As a team's NRR measures how many more runs it scores per over than it concedes, the NRRs of all the teams in a league table should sum to zero. However, because of this fact of each innings usually counting differently to the two teams' tournament NRRs, this rarely happens. If the sum is positive, this implies that overall more runs were scored per over than were conceded, which is obviously impossible. (And if the sum is negative that less were scored than conceded). The teams' tournament NRRs will all sum to zero if all the teams have played one or zero matches, or if every innings had exactly the same number of overs. This happens sometimes with small league tables. For example, Group B in the 2009 World Twenty20 featured three matches. Five of the six innings had the full complement of 20 overs, and in the sixth innings the team was bowled out, which counts as the full complement of 20 overs.

The same score by two teams counts differently to tournament NRR

If two teams make the same score from the same number of overs (either in the same match or different matches), this will count differently to their respective tournament NRR's if they have different total numbers of overs batted across the whole tournament. For example, in the 2007 World Twenty20 Group B, Australia and Zimbabwe each scored 138 from 20 overs in one of their matches. However, as Australia batted for 14.5 overs in their other match, this contributed 138/34.833 = 3.96 to their tournament NRR, whereas as Zimbabwe batted for 19.5 overs in their other match, this contributed 138/39.833 = 3.46 to their tournament NRR.

This is also the case if two teams concede the same score in the same match or different matches, but have different total numbers of overs bowled in the tournament.

Tournament NRR can penalize teams which win batting second rather than first

If one team, batting first, scores 250 from their 50 overs, and another team, batting second, is set a target of 100 which it easily reaches in 20 overs, then both sides have a batting run rate of 5. Therefore both sides will have the same match NRR, all else being equal, and should have the same contribution to tournament NRR. However, when it comes to calculating tournament NRR, the first team's innings will count more heavily than the second team's as it was longer, even though the second team achieved the same run rate and could potentially have reached the same total if it could have completed its 50 overs.

### NRR takes no account of wickets lost

In the language of Duckworth-Lewis, teams have two resources with which to score runs − overs and wickets. However, NRR takes into account only one of these − overs faced; it takes no account of wickets lost. Therefore, a team regarded as having a narrow victory can have a higher NRR than a team regarded as having a comfortable victory. For example, a team which just manages to win a close game with many overs to spare but with only one wicket in hand is likely to have a higher NRR than a team which paces itself to win comfortably with only a few overs in hand but many wickets.[3]

For example, in the 2013 Champions Trophy Group A:

• New Zealand just beat Sri Lanka by bowling them out for 138 (from 37.5 overs) then reaching 139-9 from 36.3 overs, giving them match NRR = (139/36.5) − (138/50) = 1.05.
• Sri Lanka comfortably beat England by restricting them to 293-7 from 50 overs, then reaching 297-3 from 47.1 overs, giving them match NRR = (297/47.167) − (293/50) = 0.44.
• England comfortably beat Australia by 48 runs by scoring 269-6 in 50 overs, then restricting Australia to 221-9 in 50 overs, giving them match NRR = (269/50) − (221/50) = 0.96.

The counter-argument to this is that victory is decided by whether a side has scored more runs than the opposition − the number of wickets a side takes and loses in achieving this is immaterial. Therefore judging a performance only by how quickly runs were scored could be legitimate.

### NRR may be manipulated

A team may choose to artificially reduce their margin of victory, as measured by NRR, to gain an additional advantage by not disadvantaging their opponent too much. For example, in the final round of matches in the 1999 World Cup Group B, Australia needed to beat West Indies to progress to the Super Six stage, but wanted to carry West Indies through with them to the Super Six, rather than New Zealand. This is because Australia would then have the additional points in the Super Six stage from beating West Indies in the group stage, whereas they had lost to New Zealand in the group stage. It was therefore to Australia's advantage to reduce their scoring rate and reduce their margin of victory, as measured by NRR, to minimise the negative impact of the match on West Indies' NRR, and therefore maximise West Indies' chance of going through with them.[4]

However, this is also likely to be a possibility with alternatives to NRR.

This is similar to the way a narrow victory for one side in a game of football may enable both sides to progress to the next stage, eg. West Germany v Austria in the 1982 World Cup.

## Alternatives to NRR

A number of alternatives or modifications to NRR have been suggested.

### Duckworth−Lewis

Use Tournament NRR as present, but when a side batting second successfully completes the run chase, use the Duckworth−Lewis method to predict how many runs they would have scored with a full innings. This means the calculation would be done on the basis of all innings being complete, and so would remove the criticisms of NRR penalizing teams which bat second, and NRR not taking into account wickets lost. However, this does nothing to alter the fact that when matches are rain-affected, different matches and even two complete innings in one match, can be different lengths long (in terms of overs), and so does nothing about some of the other criticisms above.

Therefore, alternatively, use Duckworth−Lewis to predict the 50-over total for every innings less than this,[5] even, for example, if a match is reduced to 40 overs each, and a side completes their 40 overs. This would make every innings in the tournament the same length, so would remove all the criticisms above. However, a side will bat differently (less conservatively) in a 40-over innings compared to a 50-over innings, and so it is quite unfair to use their 40-over total to predict how many runs they could have scored in 50 overs.

Either way, using Duckworth−Lewis would mean relying on subjective modelling predictions, which are opinions, rather than actual performances, which are facts.

### Average of the match NRRs

Calculate tournament NRR as the total or average of the individual match NRRs. This would mean all matches have equal weighting (no matter how long they were), rather than all batted overs across the tournament having equal weighting, and all bowled overs across the tournament having equal weighting. This would remove some of the criticisms above. For example, the different teams' tournament NRRs would always sum to zero if the total of the individual match NRRs were used, or if the average of the individual match NRRs were used and all teams had played the same number of games.

An example of when using this would have made a difference was the 1999 Cricket World Cup Group B. New Zealand and West Indies finished level on points. Having scored a total of 723 runs from 201 overs, and conceded 746 runs from 240.4 overs, West Indies' tournament NRR was (723/201) − (746/240.6667) = 0.50. However, New Zealand had scored 817 runs from 196.1 overs, and conceded 877 runs from 244.2 overs, so their tournament NRR was (817/196.167) − (877/244.333) = 0.58. Therefore New Zealand progressed to the Super Six stage and West Indies were eliminated. However, with individual match NRRs of −0.540, 0.295, 0.444, 5.525 and −0.530, the West Indies' average match NRR was 1.04, and with individual match NRRs of 1.225, 0.461, −0.444, −1.240 and 4.477, New Zealand's average match NRR was 0.90. Therefore West Indies' average NRR was better than New Zealand's.

### Ball difference

Ball difference (BD) is the number of balls remaining at the point of victory.[6] For a team winning batting second, BD would be the number of balls remaining. For a team winning having batted first, BD would be the number of balls between the precise delivery when the beaten team was outscored and the end of their innings (either the end of the overs or until the team were all out).[7][8] For the losing team, BD is the negative of the winning team's BD.

However, like the current NRR calculation, BD takes no account of wickets lost, so can produce similarly unjust results. In the example above from the 2013 Champions Trophy Group A, New Zealand's narrow victory over Sri Lanka would have a BD of +81, whereas Sri Lanka's comfortable victory over England would have a BD of only +17.

Also, if a match is affected by the weather, a side batting first can win having scored fewer runs, if Duckworth-Lewis increases the target for the team batting second, and they overtake the first team's score, but fail to reach the target. It's not clear what BD would be in this scenario.