DND 5e – Playable Correlation Between CR and Level

That would be misleading as there is no correlation between CR and PC level.

However, there is a guideline in page 283 of the Dungeon Master's Guide under "Monsters with Classes". It doesn't help very much, honestly, and I ended up having to:

1.) Modify a Monster

Page 273 of the DMG gives us an idea of simply taking stats from an existing creature in the Monster Manual and modifying it to our needs. This is, by far, the easiest way to make a monster. For your case, I would choose an Evil Mage Stat Block and add Sorcerer levels to it. This would have the following effects to it:

1. It gains Hit Dice of its normal type (d4 for tiny, d6 for Small, d8 for Medium and so on...) for each level you add to it.
2. It gains Sorcerer features of your choice, the benefits to this are not reliably quantified so you should judge for yourself if the class level added would increase the creature's CR.

What I found while tinkering with adding PC classes to NPCs is that for each relevant class level added, you increase the CR by 1. (I found an answer supporting this, but although it is for Pathfinder, it should give you an idea of how CR increases in relation to adding Class levels). For example, adding 3 levels of Barbarian(a relevant class) to my Bugbear NPC (CR 1) made him CR 4, pitting him and a few of his lackeys to balance out the encounter made for a fun challenge for my group.

Caveat: A sorcerer level would most likely not increase an Evil Mage's CR by a full 1, play testing and additional calculations will be needed. The reason for this is because an Evil Mage already has "Spellcasting", which reduces the "added value" of a Sorcerer level to be limited only to the Sorcerer's features until she gains more Sorcerer levels than her current spellcasting ability.

2.) Give her Legendary Actions

The downfall of solo creatures versus a party is that one side is limited to the number of actions per round it can do and in the case of the poor solo, he gets 1 Action while the players have as many Actions as they have players. This is why page 82 of the DMG raises the budgeted XP for an encounter for every creature pitted against a party. So it is a good idea to give a solo creature Legendary Actions.

Legendary Actions are a specific set of Actions a creature can take in between another creature's turn, giving more Actions (usually, 2-3 more) to the poor solo-creature.

Caveat: Legendary Actions count when calculating DPR for your OCR, adjust accordingly.

3.) Give her Legendary Resistance

Another problem for a solo creature is that it will get debuffed by a PC, and once one debuff lands, it's basically at the PC's mercy (which tends to be not a lot). Legendary Resistance gives the creature the option to succeed on a failed Saving Throw which increases its survivability against a full party with spells flinging left and right against her. Legendary Resistance increases a creature's Effective HP for each use depending on the CR as shown in DMG page 280.

When taken all together, your Evil Mage should be ready take on a party of 4 by her lonesome.

All D20, but DnD 5 especially, are designed and balanced assuming a 4 or 5 player party. And you can kinda tell, when the rules start asking you to apply multipliers to bigger parties, rather than giving any concrete guidelines. Assuming distributed player competence, a 6 person party isn't simply 20% stronger than a 5 player party. It's much higher. There's an entire additional PC, with their own suite of abilities, magic items, and most importantly, actions. Never underestimate the power of having more standard actions than the other guys (unless you're a pack of CR1/2 minions going up against a bunch of level 5 adventures, then you're screwed either way).

In short, designing challenging encounters for big parties is one of the more substantial challenges a GM may have to face. There seems to be a razor thin design space between "no actual danger" and "guaranteed TPK" when planning for large parties. My own personal strategy is "5 players per party max unless you have a super good reason that a sixth needs to be in this particular game, and never ever ever ever more than that (and preferably not 6 for long)," but that probably won't help you, specifically.

First of all, stop giving them one big thing to focus on. There are a couple iconic encounters that tend to necessitate one big monster against a party of intrepid heroes, like dragon slaying. The problem here with big parties is 1 creature generally can only attack one thing at a time, so even if the beastie is downing 1 PC a round, the rest of the party can pop cool downs and beat the timer. If you design the combat space so the beastie can use it's AOE abilites to good effect, you often find yourself looking at a TPK. Quantity may be a quality all it's own, but it isn't everything. Basically, when the party is that big, the single monster encounter HAS to be able to one round KO any given PC, or it's not a threat. And while it's killing one PC a round, the remaining party members HAVE to have a DPR high enough to whittle the beastie down before it kills them all, or they all just die. SO! Anytime you're tempted to let 7 or 8 dungeon crawlers dogpile one big boss monster, resist the urge. Instead of fighting one wyrm, why not a mated pair of adults, maybe with a wyrmling thrown in to harass the squishies? (Actually, this particular piece of design advice sort of holds true for any size party if there's more than one healer to keep the front liners standing)

Secondly (related to the first), if your party outnumbers the monsters, they'll probably win unless each monster is SIGNIFICANTLY stronger than the PC. I can't give concrete CR equivalencies because it's different at different power bands, but 7 lvl 5 PCs who know which end of a longsword is pointy should mop the floor with 5 CR7-8 monsters. The power of two players worth of extra actions is too substantial to ignore. Design the encounter using the guidelines in the DMG, then add a few support casters or bowmen(or bowwomen, or bowgoblins, or bowwhatevers) at a little under CR to bring the numbers within 1 of the party.

Finally, if you're going to do mob (mob in this context being a large angry group, not a MMO enemy, nor a crime family) encounters, consider looking at the minion style monsters from 4th edition. They had decent defenses and attack bonuses, but 1 HP and very low damage. Custom brew up something mob-able, maybe give them advantage for being adjacent to allies, with a decent attack bonus and a beefy AC, but a damage range of a d3 and very low health. And remember, mob fights aren't typically meant to be challenging in and of themselves, they should be hard enough to drain some resources while letting the players feel like badasses for steamrolling through a pile of enemies. Remember, damage spread to 7 players hurts a party much worse than that damage stacked on 1 or 2 frontliners.