How to calculate USCF's Modified Median (with or without byes)?

Question

The calculation is described in the "USCF Chess Rule Book", chapter 34E1,

34E1. Modified Median

The Median system, also known as the Harkness system for inventor Kenneth Harkness, evaluates the strength of a player’s opposition by summing the final scores of his or her opponents and then discarding the highest and lowest of these scores.

In the Modified Median system, players who tie with even scores (an even score is equal to exactly one half of the maximum possible score), have the highest- and lowest-scoring opponents’ scores excluded. The system is modified for players with non-even scores to disregard only the least significant opponents’ scores: the lowest-scoring opponent’s score is discarded for tied players with plus scores and the highest-scoring for tied players with minus scores.

For tournaments of nine or more rounds, the top two and bottom two scores are discarded for even-score ties, the bottom two scores for plus-score ties, and the top two scores for minus-score ties.

These scores are adjusted for unplayed games, which count a half point each, regardless of whether they were byes, forfeits, or simply rounds not played after an opponent withdrew. So an opponent who won the first two games, lost the third, withdrew and did not play rounds four or five would have an adjusted score of 3 points (1+1+0+0.5+0.5 = 3). These adjusted scores are used only to calculate the opponent’s tiebreaks. The player’s own score is not changed.

If the player involved in the tie has any unplayed games, they count as opponents with adjusted scores of 0.

What is unclear to me is the term: the least significant opponents’ scores. Does this include a Bye, which is a score without an actual opponent. Or is the Bye excluded from the modification procedure?

I find the text ambiguous and open to multiple interpretations, as in the answers below. Therefore, I would like to see how the calculations are performed in actual USCF tournaments.

Update-2

A cross table including the four USCF tie-breaks can be found here 2007, 128th (=129th) Annual New York State Championship. From this cross table one can conclude the unplayed games are part of the deletion process.

Update-1

Consider the 2022 US Open Championship, 30 July - 7 August, 2022· Rancho Mirage CA.

Are the default US Chess tiebreakers applied? Why is Aleksey Sorokin champion?

No.  Name                       St  Rate  Pts Rnd1  Rnd2  Rnd3  Rnd4  Rnd5  Rnd6  Rnd7  Rnd8  Rnd9  
   1  SOROKIN, ALEKSEY          TX  2648  8.0 W193  W125  W 65  W 36  W  3  D 15  D 17  W  7  W  9  
   2  MORADIABADI, ELSHAN       NC  2614  8.0 W370  W269  W 78  W 46  W 81  W 14  D  5  D  9  W 15  
   3  JACOBSON, BRANDON         NJ  2626  7.5 W281  W124  W 68  W 37  L  1  W157  W 39  D  5  W 17  
   4  EIDELMAN, GABRIEL         CA  2249  7.5 W272  W 84  W 85  L 10  W112  D 12  W 81  W 35  W 32  
   5  NARODITSKY, DANIEL A      NC  2699  7.0 W135  W 73  W 41  W 19  W 33  D 17  D  2  D  3  D 10  
   6  AKOBIAN, VARUZHAN         MO  2655  7.0 W277  W123  W 64  D 18  D  8  W 37  W 38  D 17  D 12  
   7  SHABALOV, ALEXANDER       PA  2607  7.0 W138  W 51  W 42  W 22  D 16  D 39  W 61  L  1  W 38  
   8  Hong, Andrew              CA  2604  7.0 W235  W 31  W216  W212  D  6  L  9  W108  W 24   -H-  
   9  MATVIISHEN, VIKTOR        TX  2558  7.0 W152  W126  W 70  D 15  W 72  W  8  W 16  D  2  L  1  
  10  SHENG, JOSHUA             CA  2558  7.0 W 97  W217  W 27  W  4  L 17  D 16  W 44  W 25  D  5  
  11  SCHMAKEL, SAMUEL ARTHUR   IL  2504  7.0 W140  D 76  W175  D 39  W 45  W 22  W 46  L 15  W 42  
  12  COSTELLO, ALEXANDER JOHN  CA  2379  7.0 W271  W 83  W 28  L 17  W213  D  4  W 48  W 52  D  6  
  13  MELIKIAN, GNEL            CA  2338  7.0 W 99  W167  D317  D213  W214  D 26  D 28  W 46  W 57  
  14  MEI, AUSTIN R             CA  2265  7.0  -X-  W 57  W218  D 47  W 50  L  2  W 76  D 33  W 34  

Answer 1

What is unclear to me is the term: the least significant opponents’ scores

As described in the text:

1. If the tied players have a plus score then for each tied player their lowest scoring opponent is excluded from the calculation
2. If they have a minus score then each player's highest scoring opponent is excluded.

Does this include a Bye

As you say, a bye does not involve an opponent with a score. Byes don't take part in the exclusion process.

Answer 2

It would be nice to see an example where this rule is applied. Just to be sure.

Here's a small 5 round Swiss.

 Pair | Player Name         |Total|Round|Round|Round|Round|Round|   
 Num  |                     | Pts |  1  |  2  |  3  |  4  |  5  |  
 ----------------------------------------------------------------
    1 | A                   |4.0  |W   5|W   4|D   3|D   2|W   6|  
-----------------------------------------------------------------
    2 | B                   |4.0  |W   7|D   3|W   6|D   1|W   4|  
-----------------------------------------------------------------
    3 | C                   |3.0  |W   8|D   2|D   1|L   4|W   9|
-----------------------------------------------------------------
    4 | D                   |3.0  |W   9|L   1|W   7|W   3|L   2|
-----------------------------------------------------------------
    5 | E                   |3.0  |L   1|L   6|W   9|B    |W   8|
-----------------------------------------------------------------
    6 | F                   |2.5  |H    |W   5|L   2|W   7|L   1|
-----------------------------------------------------------------
    7 | G                   |2.0  |L   2|W   8|L   4|L   6|B    |
-----------------------------------------------------------------
    8 | H                   |2.0  |L   3|L   7|B    |W   9|L   5|
-----------------------------------------------------------------
    9 | I                   |1.0  |L   4|B    |L   5|L   8|L   3|
-----------------------------------------------------------------

For player A, the opponents' scores (after adjusting for the byes) are 2.5, 3, 3, 4, and 2.5. Because player A has a plus score, the lowest score (2.5) is excluded. The modified median tiebreaker score is 12.5.

For player B, the opponents' scores are 1.5, 3, 2.5, 4, and 3. The lowest score (1.5) is excluded. The modified median tiebreaker score is 12.5. This is the same tiebreaker score as Player A.

For player G, the opponents' scores are 4, 1.5, 3, 2.5, and 0. Because this player had a minus score, the highest score (4) is excluded. The modified median tiebreaker score is 7.

For player H, the opponents' scores are 3, 1.5, 0, 0.5, and 2.5. The highest score (3) is excluded. The modified median tiebreaker score is 4.5. Player G would win this tiebreak over Player H.

Player F is not tied with anyone, but their opponents' scores are 0, 2.5, 4, 1.5, and 4. This player has an exactly even score of 2.5/5, so to calculate the tiebreak we would exclude both the lowest score (0) and the highest score (4). The modified median tiebreaker score would be 8.

What is unclear to me is the term: the least significant opponents’ scores.

The "least significant" opponent's score is the highest opponent's score for a low-scoring player, or the lowest one for a high-scoring player.

Does this include a Bye, which is a score without an actual opponent. Or is the Bye excluded from the modification procedure?

"If the player involved in the tie has any unplayed games, they count as opponents with adjusted scores of 0." The bye is not excluded from the modification procedure; it is treated like any other 0 score. In Player F's case, the 0 from the bye was the lowest score, and we threw out the lowest score, so this score was thrown out. But in the case of Player G and Player H, the highest score was thrown out, so the 0 for the bye stayed in the calculations.