Reference : "The International Chess Federation Rating System" by Arpad E. Elo (P) 1973
As explained on the description of the theory and method page, a player's performance is assumed to be normally distributed with a mean approximated by their rating and a standard deviation of 400 Elo points. The main theoretical flaw with this is covered at this link.
This assumes that the variability of all players' performance is the same, which is probably not true but is the only realistic assumption we can make without making the rating system a lot more complex and requiring a lot more data.
When we look at match & tournament scenarios, we are considering the difference of player performances. Using moment generating functions, it can easily be shown that the s.d. of the difference between two Normal distributions with the same variance has s.d. = SQRT(2) times the s.d. for each player, which is SQRT(2) * 400 = 565.7 in this case. This is (A) referred to in the table below.
If P(x) in column (C) corresponds to the probability of being at least x s.d. away from the mean on a normal distribution, x in column (A) can be found from normal distribution tables - this table actually uses MS Excel's NORMSINV function.
The Elo theory then implies that if the percentage score against opponents ranked Ry on average is 100 * P(x), column (F), then the implied Elo rating based on these games only is Ry + (S * x), where S is the s.d. of the difference between the normal distributions, i.e. 565.7. This allows us to fill in column (G) and columns (D) and (E) follow directly.
x (sds)
P(x)
TP (%)
Diff.
TP (%)
Diff.
(B)
(C)
(D)
(E)
(F)
(G)
(A) = 565.7
normal dn
= 100 - (F)
= - (G)
= (C) * 100
= (A) * (B)
0.000000
0.5000
50
0
50
0
0.025069
0.5100
49
-14
51
14
0.050154
0.5200
48
-28
52
28
0.075270
0.5300
47
-43
53
43
0.100433
0.5400
46
-57
54
57
0.125661
0.5500
45
-71
55
71
0.150969
0.5600
44
-85
56
85
0.176374
0.5700
43
-100
57
100
0.201894
0.5800
42
-114
58
114
0.227545
0.5900
41
-129
59
129
0.253347
0.6000
40
-143
60
143
0.279319
0.6100
39
-158
61
158
0.305481
0.6200
38
-173
62
173
0.331854
0.6300
37
-188
63
188
0.358459
0.6400
36
-203
64
203
0.385321
0.6500
35
-218
65
218
0.412463
0.6600
34
-233
66
233
0.439913
0.6700
33
-249
67
249
0.467699
0.6800
32
-265
68
265
0.495850
0.6900
31
-280
69
280
0.524401
0.7000
30
-297
70
297
0.553384
0.7100
29
-313
71
313
0.582841
0.7200
28
-330
72
330
0.612813
0.7300
27
-347
73
347
0.643345
0.7400
26
-364
74
364
0.674490
0.7500
25
-382
75
382
0.706302
0.7600
24
-400
76
400
0.738846
0.7700
23
-418
77
418
0.772193
0.7800
22
-437
78
437
0.806422
0.7900
21
-456
79
456
0.841621
0.8000
20
-476
80
476
0.877897
0.8100
19
-497
81
497
0.915365
0.8200
18
-518
82
518
0.954165
0.8300
17
-540
83
540
0.994457
0.8400
16
-563
84
563
1.036433
0.8500
15
-586
85
586
1.080321
0.8600
14
-611
86
611
1.126391
0.8700
13
-637
87
637
1.174988
0.8800
12
-665
88
665
1.226529
0.8900
11
-694
89
694
1.281551
0.9000
10
-725
90
725
1.340754
0.9100
9
-758
91
758
1.405074
0.9200
8
-795
92
795
1.475792
0.9300
7
-835
93
835
1.554772
0.9400
6
-880
94
880
1.644853
0.9500
5
-930
95
930
1.750686
0.9600
4
-990
96
990
1.880790
0.9700
3
-1064
97
1064
2.053748
0.9800
2
-1162
98
1162
2.326342
0.9900
1
-1316
99
1316
infinite
1.0000
0
-infinity
100
infinity
Further information:
The following links contain even more information - the less technical notes are starred:
The date and time are :
Thursday, 02-Sep-2010 21:39:56 GMT
Thursday, 02-Sep-2010 22:39:56 BST (local)
This file was last modified on Monday, 10-Mar-2003 20:05:06 GMT