Cumulative formats: stage races, series and combined events

Four things that get added

Weightlifting adds a fifth: weight. The total is the best snatch plus the best clean and jerk, in kilograms.

Adding time rewards the size of every gap. Adding places ignores it: winning by a length and winning by a minute score the same. A points table sits between the two, because its designers choose how much a better performance is worth.

Dropping the worst result

Sailing's default is that a boat's series score is the total of her race scores excluding her worst. The event's notice of race can change that: no discard, two or more discards or a number that depends on how many races are sailed.

One kind of disqualification cannot be discarded. A score marked "disqualification not excludable" always counts.

A boat that did not sail the course, retired or was disqualified scores the points for a place one more than the number of boats entered. So a bad day costs a known, capped amount, and one such day can be dropped.

Bonuses

A total can be adjusted. In cycling stage races, time bonuses exist and are counted only in the individual general classification. In cross-country skiing's stage competitions, the overall standing adds up the times from each stage together with bonus seconds and time penalties.

Starting the last event with what you have earned

The track cycling omnium is four races in one day. In the first three, the winner of each gets 40 points, second gets 38, third 36 and so on down. Riders then start the fourth, the points race, carrying those totals, and add to them or lose from them during that race through laps gained and lost and sprints won.

So the final race is the event's finish line. Whoever leads when it ends has won the omnium.

Turning the total into a head start

Some events go one step further and convert the standings into the start itself. The leader sets off first, the others follow by the margin they are behind, and the first across the line is the winner. Nobody has to do sums at the finish.

Biathlon's pursuit is the clearest case. There is no draw. Competitors start in the order in which they finished the qualifying competition, and the winner starts at time zero. Each of the others starts later by the actual time they were behind the winner.

Modern pentathlon ends the same way. Its rules say the laser run "must always be organised with a handicap start". There is one mercy for those far back: athletes more than a minute and a half behind after the earlier events all start together at one minute thirty.

Racing the first round three times

BMX racing uses addition inside what is otherwise an elimination event. Its opening phase, the motos, has three rounds. Each rider races all three, and only the combined result decides who goes on. In a category with eight riders or fewer there is nothing after the motos: the aggregate score from the three rounds is the final result.

Two matches counted as one

Team sports use addition when a tie is played over two games. In basketball's home and away series, the rules treat the two games as a single game of 80 minutes. A team that wins both goes through. If each wins one, the team with more points across the two games does. The single elimination page has the worked example.

Adding across people

A total can also be a team's. In a cycling stage race, a team's result for the day is the sum of its three best individual times. Surfing's world championships give every finishing place a number of team points, starting at 1,000 for a division winner, and add up each country's surfers. Canoe sprint's Nations Cup gives 30 points for winning a Final A, 28 for second and 27 for third, with smaller scales for Finals B and C and at least one point for every boat that took part.

You have to do all of it

Cumulative formats usually punish a gap in the record. A rider who fails to start one of the omnium's events, or abandons one, is not allowed to continue. In sailing, a boat that enters any race in a series is scored for the whole series.

Ties

With totals, exact ties are rare but the rules still cover them.

• In a cycling stage race, if riders have the same total time, the fractions of a second recorded in the individual time trials are added back in to decide the order.
• In the omnium, a tie in the final ranking is broken by the places in the final sprint of the points race.
• In a cross-country skiing stage competition, a tie in the standings during the event is broken by the number of stage wins, then second places and so on. A tie after the last stage depends on how that stage started: if it was a pursuit or mass start, the order across the line decides, and if it was an interval start the skiers share the place.

More than one table at once

A stage race keeps several classifications running. The individual and team general classifications on time are obligatory in certain events, and the team's result for a day is the sum of its three best individual times.

One sport, both forms

Golf shows the difference between cumulative and head-to-head play inside one set of rules. In stroke play every stroke over every round is added. In match play the same course is played hole by hole against one opponent, and a player wins the match on leading by more holes than remain to be played. A disastrous hole costs one hole in match play and every stroke of it in stroke play.

Lawn bowls has the same pair. In a knockout, play stops once one side cannot be caught. In tournament play the winner may be decided by the net total of shots, so every shot counts to the end.

What it is good at, and what it costs

This section is reasoning from the structure.

Adding results uses everything an entrant did, so luck on one day matters less and consistency is rewarded. Nobody goes home early. And the format can combine things that could not otherwise be compared, such as a throw and a run, by turning them into points.

The costs are subtler than a bracket's. The event can be decided before the last day, or a leader can win without winning anything. Following it takes arithmetic. And the choice of what to add is a judgement that shapes the result: the same performances can produce different winners under time, places and points.

It changes from event to event

The one thing to remember: ask three things of any leaderboard. What is being added? Is lower or higher better? Is anything left out? The answers are in the event's rules, and in sailing in the notice of race for that regatta.