Is it possible to calculate what the ELO differential is between two players based on number of games played against eachother, even if each player's ELO rating is unknown?

For example:

Player A has an unknown ELO rating

Player B also has an unknown ELO rating

However, Player A has a record of +48 =34 -18 against Player B in 100 games played against eachother.

Would we be able to tell how much stronger Player A is compared to Player B in terms of ELO points? E.g Player A is +250 ELO points compared to Player B.

How do we calculate this?

I'm not 100% positive how accurate the chart is but here is one that can answer the questions: http://www.ascotti.org/programming/chess/elo.htm

Based on your example the stronger player is approximately 262 ELO points higher rated.

But Player A hasn't won 82% of the games, he's only won 48%.  You can't include draws into a winning percentage.  So, +262 can't be right... And obviously neither can the -14 differential listed for a 48% winning percentage.  That chart doesn't factor in any draws.  Giving half a percentage for each draw seems to make sense.  48 + (34/2) = 65

Using the same chart 65% gives a differential of +110, which seems more accurate given the rather high percentage of draws. 

Does anyone know how this winning probability vs ELO rating is actually calculated?

Those draws could conceal other pertinent data.  Suppose one of these players offers draws from positions that are winning but tough and boring to play out, and that this accounted for, say, 30 of the 34 draws.  That's just one possibility of draw offer behavior affecting the ratings, and it's pretty obvious the effect would be drastically different depending on which player was charitable with offering/accepting draws.

Without doing the math, I'd assume the winning probability is calculated by determining what percentage of wins is needed for each player to keep their pre-match ratings after a match.

Source: http://www.ascotti.org/programming/chess/elo.htm

Elo rating vs. winning probabilities

This table shows the correlation between the difference in Elo points and the probability of winning a game (in bold). So for example if a player has won 60% of her games against an opponent, she would be approximately 72 Elo points stronger. Conversely, a player that is 150 points weaker than his opponent, has only a 30% chance of winning a game.

.99   +677       .66   +117       .33   -125
.98   +589       .65   +110       .32   -133
.97   +538       .64   +102       .31   -141
.96   +501       .63    +95       .30   -149
.95   +470       .62    +87       .29   -158
.94   +444       .61    +80       .28   -166
.93   +422       .60    +72       .27   -175
.92   +401       .59    +65       .26   -184
.91   +383       .58    +57       .25   -193
.90   +366       .57    +50       .24   -202
.89   +351       .56    +43       .23   -211
.88   +335       .55    +36       .22   -220
.87   +322       .54    +29       .21   -230
.86   +309       .53    +21       .20   -240
.85   +296       .52    +14       .19   -251
.84   +284       .51     +7       .18   -262
.83   +273       .50      0       .17   -273
.82   +262       .49     -7       .16   -284
.81   +251       .48    -14       .15   -296
.80   +240       .47    -21       .14   -309
.79   +230       .46    -29       .13   -322
.78   +220       .45    -36       .12   -335
.77   +211       .44    -43       .11   -351
.76   +202       .43    -50       .10   -366
.75   +193       .42    -57       .09   -383
.74   +184       .41    -65       .08   -401
.73   +175       .40    -72       .07   -422
.72   +166       .39    -80       .06   -444
.71   +158       .38    -87       .05   -470
.70   +149       .37    -95       .04   -501
.69   +141       .36   -102       .03   -538
.68   +133       .35   -110       .02   -589

The number of draws are irrelevant. All that matters is the winning %. There are potential complications if the players' ratings are over 2100 because there's something called a "k" factor. But this is the basic, simple formula that the rating system is based on.

So everyone who gave 65% as being 110 rating points higher was correct.

[Edited to correct a grammatical mistake.]

By my calculations, they are about 230 ELO points apart. A player 200 points below in elo is expected to score 25%. Thus, a player 230 points below will score 21%, as in your example (this approximate number is also given by Smyslov fan for a score of 21%).

But draws do factor into it, they are included in the calculation as far as I am aware (when I draw my ELO changes). Therefore I calculated the percentage as:

100 * (Wins + Draws / 2) / Total games

You don't add the differences! The difference is the same whether it's 65% wins or 35% wins. It's not 220 points difference, it's 110 points difference!

My bad, miscalculated, I'm missing 17 games in my calculation.

Player B scored 35 points. There were 100 total games, so he did score 65%.

I counted the draws as only half games on accident, which destroyed my calculation. They are half points, but full games.

I agree with SmyslovFan and Estragon here.  I brought up the effects of draw offer behavior here, however, because it seems the OP might want to draw conclusions about real people from a real set of games, and his rating doesn't line up with such a high percentage of draws.  Neither does mine, for that matter.

While I know GMs can make mistakes in offering or accepting draws, I'm far more inclined to trust their judgment.  I don't expect class players to have a significant percentage of draws, though, so it seems to me that charity or lack of trust in endgame skills might cause a class player to offer/accept draws that should have been wins.  Been there, done that.

If this thread represents an effort to draw rating conclusions about real class players and a real set of games, it's worth considering this angle.  Otherwise, I'm just blabbing and wasting my time.  

Fingerly, imagine these two sets of results:

5 wins, 2 losses, 3 draws (65%)

3 wins, 0 losses, 7 draws (65%)

The rating difference in BOTH cases is 110 points. It doesn't matter how you arrived at that figure. 

The amount of draws is completely irrelevant!

I understand the math, and I'm not disputing it.  It's the quality of draws at the class level that I'm talking about: a large percentage of draws between two players at the class level indicates something is amiss to me, and the likely X factor seems to be something outside of the games themselves.  I'd expect one of those players is giving up on slightly-better games as draws for one reason or another.  If so, it skews that player's  rating.  

Ultimately we can only base ratings on end results, making this point moot--but a desire to draw conclusions about two class players in this scenario can't be complete without asking why so many draws are taking place.  In the OP's example, if Player B was arriving at slightly-better positions and then offering draws because the ending would be tough and boring in 30 of those draws, and if Player B had played those games out instead and won them, the final tally would be +48 =4 -48.  It would also more-closely resemble a plausible set of results from class players in OTB play.

OK, I didn't think that through too much when I posted the link to the chart. I was basing my thought on the fact black won 18% of the games played, which doesn't make sense as the numbers should match when looking at the white win percent too.

So, since draws are ignored, then the percents we would need to look at would be based on the decisive games. In the example given, there are 66 decisive games with white winning 48 and black 18. Wouldn't that give .73 an .27 (rounded) and give a difference of 175 ELO?

I think a player rated 100 points higher would win 5/8 so by checking the points it should be possible.

Just read through an article on ELO calculations and draws are considered half win, half loss. So white is 65% and black 35%. Learn something new every day.

My question is not about ELO or whatever. I simply want to know how to determine the winning percentage from the total games played. My question is, for example, if I played 82 games out of which I won 63, lost 12 and drew 7 -- what would then be my winning percentage and how do you calculate that? Thanks for your support.  

Ag, treat draws as half points and add them to the win column, then divide that by the total.

63 +3.5 =66.5/82 = about 81%

Thank you so much SmyslovFan! Your reply is perfectly clear.