Are ratings of players who do 10 minute games equivalent to players who play 15+10 games?

For example would a 1200 player who plays mostly 10 minute games be the equivalent of a 1200 player who plays mostly 15+10 games? If they are playing in different pools then the ratings cannot be compared directly, as a rating is only relative to the pool a player is playing in. 

The reason why I am asking is because the player pool for the different time controls may be significantly different, I'm pretty sure far more people play 10 minute games than 15+10. Assuming players mostly stick to their preferred time control. 

So would the 1200 player who plays 10 minute games be stronger or weaker than a 1200 15+10 player?

I've been wondering about this for a while. For the rapid rated games, I almost always play 15 | 10. When I occasionally play 10 | 0, I will most likely lose because of the time trouble or just because the opponent plays better. It would be fun if chess.com publishes some stats on how well the players perform in time controls that are slightly different from their favourite time controls.

No they are not the same.
The player pools are different.
It is also a different game. What works in 10+0 does not work in 15+10.

I would guess that the 15+ rating is more accurate of a player's ability.

10 minute has the biggest pool of players of all time controls.

If the players pools are different then the rating are not equivalent. For example if better players tend to go for 15/10 games then 1100 rating of player who plays mostly 15/10 games will indicate that he is better than 1100 player who only plays 10 minute games. 

what is important is not the size of the pool but whether players in that pool are on average better or worse than in the other pool.

I play 10 minute for my rapid 1670 pretty much exclusively, I might try playing 15/10 for a couple of months, my guess is my rating will go down maybe 200 points.

That is exactly why I think they should add longer time control rating.  

The biggest difference is between 10|0 and some long time control like 45|45 or 1 hour per side. 

I found 15|10 slightly easier, and rose a little bit in rank. But I might have gotten paired against particularly weak opponents at first. Probably also has to do with my play style. I just really cannot stand 10|0 anymore, losing on time when I have a winning position.

I believe there used to be a "standard" rating, though they removed it.

Good point, the more time I have the higher my rating is in the different time controls.

     The real question is: would someone who plays almost exclusively 10 min games be weaker, stronger, or equal to someone who has a near-identical rating but always plays 15/10 games in their overall chess understanding and ability? 

     My feeling is that the player who has played the longer games will be better. Just having spent more time analyzing the positions will probably teach the 15/10 player a bit more about the game. Also, the quality of play will be a bit higher in the longer games, meaning the 15/10 player has to be a bit stronger to obtain the same rating.

     Were the two to play a blitz game the 10/0 player would be more likely to prevail, but in a classical otb encounter I'd bet on the 15/10 player. Of course when talking about 1200-rated players the number and severity of mistakes will be the deciding factor.

It probably depends on what time control 10|0 person and 15|10 person decide to play.

For instance, I am pretty sure if I played someone my rating who plays 10|0 exclusively that I wouldn't fare well in 10|0 at least in the beginning (I mean I wouldn't play my best until I get used to time control). In the same way if we were to play 1 hour per side, I am pretty sure 10|0 person would play too fast until he/she gets used to that time control.

10|0 and 15|10 is a less extreme example but it is different as well. But who is objectively stronger in chess understanding between the two, who knows, it is probably different from case to case. For instance, some 10|0 player might play some tricky openings that are objectively not the best but will make their opponent flag, while others might play something sound. The second player might have more chess understanding but the first one might still be higher rated.  

That is why I am not sure we can have some general conclusion based on a few players, so drawing conclusions might be a bit difficult. 

I personally don't really feel a difference when I switch from 10|0 to 15|10 and vice versa so imo there's probably not a huge difference (But I do usually play 10|0 so maybe I'm biased)

It is not a huge difference, but it is still different.  For instance, if you get to the endgame, 15+10 is pretty close to 30|0 (you have 25 minutes if you get to move 60 for instance). For a normal game you will have twice as much time in 15+10 so it is not a small difference either.

I think that you are probably right. 10|0 is a treacherous time control. You think that you have some time, and in reality you don't. In some capacity it might be easier to play 5|3 than 10|0 for un unexperienced player. 

In any case 10|0 is still a speed chess for sure, at least for most of us.

No, I dont think so as I always somehow lose on time in 3|0 and was stuck at lower rating but have a nice win% in 5|5 and increased my rating by 300+ and still increasing but still both are blitz, the time controls best suite from person to person, so every different time control may have different rating of any person.

If one is interested enough one could play numerous games at each time control and see if the results are comparable.  

It probably differs across different ratings as well. For instance, a few months ago, I played a game against the opponent rated around 1350 in 1 hour per side game.

We played Ruy Lopez and he tried fishing hook trap, but it just didn't work in that particular position. I took around 4-5 minutes to try to calculate what will happen if I take the [piece because it felt that he will be lost. I haven't seen everything for sure, but I was right that it just didn't work, I found the right knight move afterwards and I won easily. Apart from that single trap, he played really poorly for that rating. I saw that he played multiple 10 minute and blitz games with the same trap and of course he made it worked here and there. 

If we've played a 10 minute game, I could've flagged in that same position as I would take some time for sure.

Of course, someone rated say 2200 who plays blitz or 10|0 who plays trappy openings will play in a much more sophisticated way and in that case I am sure such a player might have a better chess understanding. But I think that 10|0 is very fast and deceiving time control for most players here, and someone who learns some traps can be at least partially successful, at least on lower to intermediate levels.