http://imgur.com/e6HUxX3<- super tl;dr

SKIP INTRODUCTION IF YOU WANT TO JUMP STRAIGHT TO THE EASY TO LOOK AT STUFF.
SKIP INTRODUCTION IF YOU WANT TO JUMP STRAIGHT TO THE EASY TO LOOK AT STUFF.
SKIP INTRODUCTION IF YOU WANT TO JUMP STRAIGHT TO THE EASY TO LOOK AT STUFF.
SKIP INTRODUCTION IF YOU WANT TO JUMP STRAIGHT TO THE EASY TO LOOK AT STUFF.

tl;dr Used some algorithms to determine what champs you should main if you want to climb to the highest of elos.

Warnings:

1) Based on Korean SoloQ data, might be different in other regions. However, the KR soloQ is considered to be the most competitive in the world. (in future, might introduce data from the other regions)

2) Subject to change across different metas and patches - the data can be considered long-term data. Hence, for example, if a champion has been heavily nerfed in a very recent patch, the champ might not be as effective as shown in the graph below - it is better to think of this as a season-by-season or 'quarterly-by-quarterly' based 'best champs to main' analysis, than a 'week-to-week' based 'best champs to main' analysis.

3) The data doesn't necessarily consider what is being played right at the absolute top of the Korean ladder* - e.g. top 10, top 30, top 50 (~50 players in total) - instead it's focused on what is played by champion mains D5-Challenger (~50,000 players in total), and what champions allow players to climb to the higher ranges of Masters & Challenger.

* Might add some data regarding this, but this analysis is meant to be more general. Also from what I've casually looked at you won't find too much divergence from my data - the same power picks are regularly found.

4. There's differences between SoloQ & competitive play, some champs are a lot more effective with team synergy and communication for example, than without. In a competitive 5 man team game, there will surely be many differences in champion carrying ability. This analysis is mainly intended for a 'soloQ' (RIP) playstyle.

Before looking at the analysis for all champions, I want to assert that not all champions are good across all elos.

There is a huge difference between Challenger & Master and the rest of diamond. This can be seen pretty easily from this graph that showns the distribution of high elo players in Korean soloQ. Challengers are ~0.4% of all high elo players.

http://imgur.com/C9EMvbL
You could think of it as if, the difference between Bronze & Silvers and Diamonds is the same as the difference between low Diamonds and Masters & Challengers (not exact, but you get the point).

Strategies and champion picks change in effectiveness across elo. Strategies that might be totally okay and competitive on average in Bronze - such as AP Yasuo, or AD Jungle Soraka - won't be as effective in Platinum level games, while they'll be totally and absolutely useless on average in high elo games.

http://imgur.com/tJhVADZ

Based on this sort of reasoning, if you wish to acquire elo and main champions that are competitive and good at carrying at very high elo, you should pick champions that are shown to be performant by many players already on the ladder already.

My analysis suggests that a few champions are highly effective at carrying yourself to very high elos, while most champions are mediocre.

All champions (up to Aurelion Sol),

http://imgur.com/birHcju

By champion (up to Aurelion Sol),

http://imgur.com/e6HUxX3

Warnings:
- Some champions found lower in the chart might still have 1 or 2 anomalies (i.e. mains) that push the champion to the limit and are able to play them at much higher elos, such as someone able to play Garen at master elo level for example. On average however, given a champions playerbase, most aren't able to do the same on average. This essentially means, you can still play champions that are lower in my chart if you truly wish too. This is just a guide for lazy & desperate people who wish to acquire elo.

TODO
- Analysis of champs by the absolute top 50/top 100 players on the ladder
- Add data for Taliyah when she is released
- Finish analysis of top 15: #6 onwards
- Analysis of Top 5 per Role

Competitive Score places more value on champions that are particularly effective at Master & Challenger level, and places much less value on those that are only effective at Diamond.

http://imgur.com/e6HUxX3

http://imgur.com/1tMmBUM
The champions with the highest ability to carry themselves at very high elo are naturally a few junglers that are extremely competitive relative to their counterparts.

This is because for much of the game laners will be constrained to exerting pressure in their lanes since that's where all their farm (i.e. gold/levels etc) is, however a jungler on the other hand can exert pressure anywhere on the map right from the start since their farm is more spread out.

If you are very skilled on these few jungle champions, your ability to tilt games will be massive allowing you to acquire elo.

The 4th ranked champion, Bard, a support, is interestingly powerful for carrying yourself at super high elos arguably for the same reasons - since Bard is encouraged to roam given solely it's source of xp chimes and a kit that has tons of mobility, CC & game changing ulti, skilled Bard mains can also impact the entire map with roams.

http://imgur.com/cybVKhG

Looking at the top 25, it seems the roles with champs with high carry abilities are mid and jungle.

Jungle has partly been explained, but it's unsurprising for mid to be so high also.

Top laners, especially without teleport, can likely only impact top lane and mid lane (by roaming - since mid is pretty close, bot lane is very very far - hence you lose out on farm heavily if you roam bot) for most of the game.
Bot laners, can likely only impact bot lane and mid lane (by again roaming) for most of the game.
Mid laners however, have more ability to impact top, mid and bot since it's relatively close enough to all lanes and can roam without losing much farm.

Conclusions:

To acquire elo,
- Main high pressure junglers
- Main supports incentivised to roam
- Main jungle or mid since they are the roles with the most ability to carry themselves by having the ability to impact all the lanes

#1 Elise

Heavy pressure jungler.
Heavy sustain in the jungle.
Can gank early.
Ganks have high success rates relative to other junglers (esp with red buff since ranged).
Strong duelist.
Strong ability to safely tower dive - even early in the game.
High damage that doesn't really fall off given execute even when built reasonably tanky.
Mobile & partial invulnerability.
Powerful CC for picking off champions late game, the CC allows her to be relevant in late game team fights - and can often be enough to engage.
Teams will always have AD since adcs exist, but many teams can lack AP - Elise can offer that.

IF skilled, very powerful at all stages of the game.

She was said to be the result of Riot intentionally trying to create an AP champion that worked in the jungle, and they succeeded.

No other jungler can tower dive as effectively as her pre-6, this means for a significant proportion of the game she can pull of ganks no other jungler effectively can. When enemy laners will think they are safe under turret, they won't necessarily be if a skilled Elise is in the game.

Few junglers can also pick off enemy champions as well as her also, as her whole combo especially when fed, can result in fast solo-kills. This is powerful as deleting a champion of the map creates a lot of pressure.

Specialities i.e. some things you should focus on practising:
- Tower diving, particularly early in the game
- Vision Control - hide in bush - then utilise your CC to get picks.
- Red buff ganks

#2 Rek'sai

Heavy pressure jungler.
Heavy sustain in the jungle.
Can gank early.
Ganks have high success rates relative to other junglers (unavoidable Flash engage).
Can gank from uncommon and unpopular routes (with her e)
Strong duelist.
Ulti provides an incredible amount of map presence.
Mobile with both her ultimate and her tunnels (tunnel placement shouldn't be underestimated).
Strong CC for engaging, and strong bruiser/tank hence effective at engaging.

IF skilled, very powerful at all stages of the game.

She is another jungler specifically designed to work effectively in the jungle - in particular with her own niche (map presence with her ultimate).

In particular, her ultimate provides an extreme amount of presence across the map. Her ganks aren't the best hence it is often better to farm with her - ideally in the enemy jungle -, however she can still pull off high success rate ganks with her flash knockup.

Specialities i.e. some things you should focus on practising:
- Reading the map (enemy jungler presence to counter jungle, tunnel locations)
- Ganking from alternative routes
- Counter Jungling
- Engaging esp with Flash

#3 Gragas

Heavy pressure jungler.
Heavy sustain in the jungle.
Can gank early.
Ganks have high success rates relative to other junglers (unavoidable Flash engage).
Pretty strong duelist.
CC heavy & builds tanky - so a powerful team fighter and skirmisher.
Mobile with pretty low cd dash (esp if hits enemy).
Powerful engager/disingager.

IF skilled, very powerful at all stages of the game.

There's reasons why champs like Janna are very powerful in soloQ, effective disengaging and engaging are extremely valuable skills for building well balanced/powerful team comps.

The ability to do that while having tons of CC and being able to build tanky, while also have respectable damage, means Gragas is a highly effective pressure jungler and team fighter in the right hands.

His early game ganking and tankiness isn't the best, but his kit still has high success rate ganks with his flash E.

Specialities i.e. some things you should focus on practising:
- Reading the map (skirmishes - protecting allies by disengaging or engaging on free kills)
- Flash E ganks
- Utilising ultimate effectively to both engage and disengage fights and skirmishes

#4 Bard

The defacto roaming support - can roam without losing out much since he gains xp via chimes.
High pressure support.
Powerful difficult to dodge CC - his auto-attacks can slow also.
Ability to pull off uncommon ganks - routes, bring other ppl along etc etc.
His ultimate is extremely powerful for picking off poorly positioned enemies (will occur consistently against weaker opponents) that can win games by itself
High success rate low-risk engager/disengager - multi stuns during teamfights from long range are incredibly powerful and can occur often with low cds.
Decent laning phase with lane sustain, and lane poke.

IF skilled, incredibly high playmaking potential.

Every team comp needs play makers, every team comp needs champs with low risk high success rate CC and the like to both get to their carries and shut them down.

It is fair to say the champ with the highest playmaking potential in the game currently is Bard, and it's not suprising he's ranked #4.

Lower elo you might see a lot of unskilled bards that often seem to help the enemy team instead and hence twists your view of the champion, and makes it appear that he's a lot more risky than he is, but in the right hands you'll find a lot of his kit is actually very low risk high success - this requires skills you'll only find higher elo - positioning ability, prediction ability, map awareness, knowledge of when to roam and how to roam etc etc.

Specialities i.e. some things you should focus on practising:
- How to roam, when to roam, what routes to take etc etc
- Landing your Q
- Utilising magical journey to gank evading enemy vision - ideally with your jungler
- Recognising when your ultimate can be used to both engage and disengage

#5 Lucian

* as of writing this, there's plans to nerf him, so depending on this a lot can change - however, he's been nerfed heavily in the past too. He always re-emerges as THE defacto adc however, after all the other adcs have been nerfed (it's a cyclical thing).

The defacto adc that has a kit that is not the best at all stages of the game, but still strong at all stages of the game.
High mobility, with the ability to run a diverse set of builds - occasions when almost all other adcs would die, he won't.
High impact ultimate allows him both waveclear, and deal a lot of safe damage - and also gives him strong dueling potential.
High pressure - can consistently output dps safely, and reposition himself, hence very powerful under skilled players.

IF skilled, consistent & reliable.

Every team comp needs a consistent reliable adc to output damage, hence why across all metas, adcs are always picked, Lucian is that adc - when he's not been over-nerfed at least.

Other adcs have their niches, and even right now are arguably more impactful picks especially in a 5 man team based setting, but Lucian is the adc that will rarely ever let you down.

Don't overfocus on his reasonable win rates across elos also, since he retains such win rates while usually being the most played adc (often champion) in the game - hence suffers from regression to the mean from all the less skilled players playing him.

Specialities i.e. some things you should focus on practising:
- Generic adc things: farming, positioning, item builds etc etc
- Roaming (remember he is incredibly mobile hence can gank quite effectively) and is also a safe roamer given his ability to dash over walls
- Repositioning himself and utilising passive