Median-Buchholz: Swiss tiebreak

Median-Buchholz

Definition

Median-Buchholz is a Swiss-system tiebreak method derived from the Buchholz score. Like Buchholz, it adds up the final scores of a player’s opponents, but it reduces the influence of extreme results by discarding the highest and the lowest opponent scores before summing. The idea is to measure the “typical” strength of a player’s opposition, minimizing luck from facing an unusually strong or weak outlier.

How it is calculated

1. List the final scores (points) of all your opponents at the end of the tournament.
2. Order these scores from lowest to highest.
3. Discard the lowest and the highest opponent scores.
4. Sum the remaining scores. This sum is your Median-Buchholz tiebreak.

Common variations organizers may specify:

• Median-Buchholz 1 (often just “Median”): drop one highest and one lowest score.
• Median-Buchholz 2: drop the two highest and two lowest scores.
• Buchholz Cut 1: drop only the single lowest score (this is not Median-Buchholz but is frequently listed alongside it).
• Modified Median: a federation-specific tweak that adjusts which extremes are dropped for players with extreme totals (e.g., perfect or zero scores). Exact rules vary; always check the event regulations.

Usage in chess

Median-Buchholz is widely used to break ties on the same number of points in Swiss tournaments—rapid, blitz, and classical. Many events list it near the top of their tiebreak order because it strives to reward consistent performance against a representative field, not distorted by one outlier opponent.

Typical tiebreak orders might include combinations such as direct encounter, Median-Buchholz (or Buchholz Cut 1), full Buchholz, Sonneborn–Berger, number of wins, or rating performance. The exact order is event-specific and published in the tournament’s regulations.

Strategic significance

• Tournament context: While you cannot control who your earlier opponents play later, your Median-Buchholz improves when your opponents score well overall—so players often find themselves quietly rooting for their past opponents.
• Last-round decisions: If standings are tight, players sometimes assess their likely tiebreaks to decide whether a draw suffices or a win is needed. A solid Median-Buchholz can make a draw more attractive; a weak one might compel risk.
• Fairness aim: By discarding extremes, Median-Buchholz reduces the “luck” factor of having faced one unusually strong or weak opponent.

Examples and calculations

Example 1 (5-round Swiss): A player scores 4/5. The final scores of their five opponents are 2.0, 2.5, 3.0, 4.0, and 5.0.

• Buchholz (full): 2.0 + 2.5 + 3.0 + 4.0 + 5.0 = 16.5
• Median-Buchholz (drop highest and lowest): discard 2.0 and 5.0 → 2.5 + 3.0 + 4.0 = 9.5
• Buchholz Cut 1 (drop lowest only): discard 2.0 → 2.5 + 3.0 + 4.0 + 5.0 = 14.5
• Median-Buchholz 2 (drop the two highest and two lowest): discard 2.0, 2.5, 4.0, 5.0 → remaining 3.0

Example 2 (illustrating robustness): Two players tie on 4/5.

• Player A’s opponents’ final scores: 1.0, 2.0, 3.0, 4.0, 5.0 → Median-Buchholz = 2.0 + 3.0 + 4.0 = 9.0
• Player B’s opponents’ final scores: 2.5, 3.0, 3.0, 3.0, 3.5 → Median-Buchholz = 3.0 + 3.0 + 3.0 = 9.0

Note how Player A’s extreme 1.0 and 5.0 opponents are removed; both players end up with the same median measure of opposition, despite A’s larger spread.

Historical notes

Buchholz-based systems are named after Bruno Buchholz, who proposed the original method of summing opponents’ scores as a tiebreak. The “Median” refinement came later to reduce sensitivity to extreme pairings, and it gained popularity as Swiss opens grew larger and more rating-diverse. Many national federations and FIDE-rated events include Median-Buchholz or a “Cut” Buchholz in their published tiebreak orders.

Edge cases and implementation details

• Unplayed games/byes/forfeits: Events specify whether byes or forfeits count as opponents with a particular score, or are omitted/replaced. Check the regulations.
• Small numbers of rounds: With very short events (e.g., 3 rounds), dropping extremes can leave little data; organizers may choose Buchholz Cut 1 or no cut in such cases.
• Perfect or zero scores: Some “Modified Median” rules adjust which scores are dropped so that players with 5/5 or 0/5 are not disproportionately helped or hurt by discarding an extreme.

Related terms

• Buchholz: The base system summing all opponent scores without discarding extremes.
• Buchholz Cut 1: A variant that drops only the lowest opponent score.
• Sonneborn–Berger: A different tiebreak that weights opponent scores by the results against them (common in round-robins, sometimes used in Swiss).
• Modified Median: A federation-specific adjustment to Median-Buchholz, often for extreme total scores.

Interesting facts

• Players sometimes track their “tiebreak race” in the final round—watching prior opponents’ results closely, since each extra half-point they score can swing the Buchholz and Median-Buchholz columns.
• Two players can have identical Buchholz but different Median-Buchholz if one faced a single very strong (or weak) outlier and the other had a steadier schedule.
• Median-Buchholz tends to correlate with the “strength” of your path through the event, but it is not a rating-based metric; it uses results only.

Practical tips

• As a player: After each round, note your opponents’ running totals. Before the last round, estimate whether a draw might hold on tiebreaks.
• As an organizer: Publish exactly which variant you’ll use (e.g., “Median-Buchholz 1” or “Buchholz Cut 1”), and how unplayed games and byes are treated, so players can compute tiebreaks transparently.