It is hard to get your head round because lower Ingo ratings are better than high ones (opposite of modern Elo based systems) and because none of the references above give numbers. So, what would Magnus Carlsen's Ingo rating be? What would my Ingo rating be? What about an "average" player? What about a beginner?
The Wikipedia reference gives some details of how the calculations work but misses out key details that still leave a feeling of incomprehension:
New players receive a high, fixed starting score. Players' new ratings centre on the average rating of entrants to their competition: then if having achieved better than a net draw set of result, minus the number of percentage points it is over 50% (e.g. a 12–4 or 24–8 wins-to-losses result is, as ever, noted as a 75% tournament outcome) – if having achieved worse than this then the number, again in percent, is added to the average of the tournament entrants' scores; thus in all cases recalibrating all players after each tournament completely. A consequence is at most 50 points gained or shed per tournament (namely by a totally winning or totally losing participant) away from the tournament average.
This begs a number of questions.
What is the "high, fixed starting score" given to new players?
Does a round robin tournament with say 4 players really have the same weighting as a round robin with say 10 players? Seems unfair if true.
I found a number of references which clear up some of these questions.
First, this chess.com reference gives a formula for converting between Ingo and FIDE and also notes that Ingo ratings typically vary between 0 (strong, world champion) and 300 (complete beginner):
FIDE = 2840 - (INGO x 8) and
INGO = (2840 - FIDE) / 8
That would put Carlsen round about 0, an average 1500 FIDE player would be about 170 and a player who just gets a 1000 FIDE rating would be about 230.
The Ingo rating system may no longer be used for classical chess ratings but it is used for the Chinese variant, Xiangqi, at least by the European Xiangqi federation. I found this useful reference for its use in Xiangqi which gives more details.
1. Principles: a) The "better" player the lower rating.
INGO range is from about 10 (world champion) up to 250 (beginner), maximum rating should be 300.
b) Rating performance has to be calculated strictly in chronological order of the tournaments.
c) The ratings will be updated after each tournament.
d) Each INGO rating gets an index that means the number of tournaments that have been calculated for this player.
Example: a rating of 120-10 means an INGO of 120, resulting out of 10 rated tournaments.
2. Meanings: One point difference means one percent more or less of expectancy range.
Example: If A has a rating of 110 and the average of A's opponents is also 110, the expectancy range is exactly 50%; if the opponents' average is 100, expectancy range is 10% less than 50%, means 40% (when playing 10 games A should gain 4 points otherwise his INGO rating has to be changed).
3. Limits: Due to the linearity of the INGO system, the calculating of expectancy ranges is limited to 50 points difference. (Of course it is not possible that a player may gain more than one point resp. less than 0 points per game..) So we have to cut off in case of opponents' rating differs more than 50 points from yours, when calculating your opponents' average.
Example: A's INGO is 100, his opponents' 100,170,40, for calculation of average take 100,150,50, so the average will be 100 and A should gain 50% that means 1.5 points of the three games (three points in "Chinese" counting style).
4. Performance: Row "H" shows the rating performance of a player for the calculated tournament.
5. New INGO: Abbreviation of INGO is "F" (F-old, F-new).
F is normally calculated as follows:
F-new = (F-old * k + H) / (k + 1), where k means the expansion coefficient:
6. Coefficient: The lower coefficient, the faster the changes!
Normally the expansion coefficient is
k = 3; if your index is less than 10, make:
k = k - 1; if your index is less than 5, make:
k = k - 1; for every game less than 5 in the current tournament, make:
k = k + 1; if the tournament's time schedule is less than 1 hour per player (quickplay), k will be doubled:
k = k + k but see 7.a Restrictions)
Example: your rating is 120-4 and you play 3 games in your 5th tournament, your coefficient is:
k = 3 (starting value)
-1 (because less than 10 tournaments)
-1 (because less than 5 tournaments)
+2 (2 games less than 5)
If this tournament is a quickplay tournament, k will now be doubled to 6!
It looks to me like this Xiangqi variant of the Ingo rating system has been updated to modern requirements (quickplay, handling weaker and stronger players with "k" factor, etc.) but it makes how it works much clearer than the Wikipedia article.