Mug little old ladies with black belts? I draw the line at brown. 
I would say from experience, that dojo freesparing goes out the window in a real street fight.
I was in high school when I got jumped by some guy who looked liked he was in his mid to late 20’s. He thought my friends and I had egged his car on Halloween and wanted a piece of our ass.
I had a couple of years of Tae Kwon Do training and had fought in tournaments before.
When he wound up to punch me from 10 feet away, I was expecting it to be a feint—he didn’t really think he could hit me with that punch, did he?
But he did, and I leaned back just enough so his fist just passed my face; and as he missed his whole head was wide open and I threw a hook at his temple—this is what I would have done in the dojo and was thinking to myself “goodbye dumbshit”—and I missed!
Within seconds it became a grappling struggle on the ground.
I was lucky that I had wrestled in high school and this guy obviously hadn’t. I got the upper hand (he didn’t find the ¾ Nelson too comfortable) and he gave up and got in his car and left.
Bottom line as I see it and as posters above have stated is to keep a cool head. And I do think that the dojo training gave me that. A lot of time it’s the mentally superior person—not the “physically” superior person that prevails.
Not. He was a sumo guy, never was a fighter in the sense we are talking here. Plus he’s super fat. Not a representative example of the OP I’d argue. Gracie, otoh, is a great example of a small grappler who can best bigger opponents.
Yes. I have had one student in my school who was attacked by two men, both of whom were more than a foot taller than she was. Both of them went to the hospital before they want to jail. One of them lost an eye.
I’m about 5’10", my 11-year-old neice was about 4’10" when she floored me with an elbow smash to my armpit.
Maybe I shouldn’t have taught her that one.
First of all I’m going to assume that the OP wants to know about actual, visceral, -I have to disable this person because I don’t know him/he might kill me - fights.
If that’s true, then the OP has to define what he means with “martial arts”. Many modern martial arts are sports, and not meant to help you in such an encounter, though many people will (wrongly) tell you otherwise. This includes TKD, much of the Karate you’ll see done in Dojos, and the list goes on and on.
The best advantage such sports will give you is your physical condition. If you’re doing these types of martial sports you’re likely to be in good enough physical condition to get out of the situtation by forcibly finding a way out and out running your opponent. This is probably the best thing to do anyway.
People who actually used combatives to defend themselves (either back in the day, or modern people unlucky enough to actually have to defend themselves without guns - few and far between) will tell you that the larger, stronger opponent will have the clear advantage and only superior skill will win the day.
I don’t think we can establish a magic ratio however, every situation is unique. The strong, big guy can make a stupid mistake, or perhaps the smaller guy will simply fail to overcome the disadvantages.
So while we can say that size and strength prsent an advantage, we can’t quantify it, nor can we quantify relative skill levels.
I think you need to make a distinction between someone with some fighting training, and someone with considerable expertise. Even a “black belt” is a far cry from someone with true skill and years of experience.
For example, my old instructor’s wife could easily kick my ass - especially if weapons were involved or if we went to the ground. I’m 6’3" 200#, she was around 5’5", 140#. I could probably hold my own boxing or kickboxing with her, which is why she would never box/kickbox me.
But she trained and taught - primarily with men - nearly all day every day.
IMO, the vast majority of her students, while they gained some significant skills, would have difficulty against a committed, significantly larger opponent.
Also, your OP specified karate. It is quite common for individuals to get advanced belts in karate (and other arts) with little or no experience fighting. Your prettiest kata woun’t do you much good in a real fight. And unlike kumite, fights aren’t stopped after the first touch.
To answer your question, I think it would be an exceptional unarmed woman who could defend herself against a committed attacker nearly twice her weight. The main thing training can get a woman is awareness to avoid a confrontation, and surprise, time, possibility of escape, and to dissuade an attacker to go after a less troublesome prey.
When I started fighting NHB bouts, I was 185#, right at the bottom of the heavy weight category in most fights. Accordingly, I was going up against guys as large as 300#, with roughly similar skill/experience level as me. Once they got more than 30-50# heavier than me, it was a big advantage - especially if they got on top of you on the ground. Also, most big figthers punch really hard, and are able to absorb some damage.
Royce, and Rickson, are pretty exceptional athletes. But look how quickly the UFC got dominated by huge wrestlers.
There’s nothing magical about martial arts that will allow you to beat up a skilled opponent much larger than you. Wrestling and boxing are martial arts. There’s a reason they have weight classes.
As I said in another thread, if you are an out of shape little guy, I don’t believe there is any “trick” you can pull to beat a bigger, more skilled, better conditioned fighter. What “trick” would you teach someone to beat an NBA basketball player in a one on one game?
A smaller opponent CAN beat a larger one if he is better conditioned, better trained and more highly skilled though.
Nothing, let me repeat, Nothing, nada, zilch, zero beats mean and likes pain*.
*or, if the difference matters, doesn’t care about pain.
A tae-kwon-do teacher I knew once said he refused to teach to smaller women as a form of self-defense - he said that there were other self-defense classes that would be more useful as they were more about incapacitating or distracting your attacker and getting away, and what he taught was fighting. I can’t remember the exact belt (I know the black belt was not the highest example he gave so I think it’s graded differently than karate) but he said that you needed a hell of a lot more training (several belts) to have a good chance to beat a fighter who was 150% your size, and it was damn near impossible for someone to be able to beat someone twice their size (this is assuming that the opponent is reasonably fit, I’m sure he’d have no trouble beating a morbidly obese man twice his weight). Anyway, for most women their attackers are usually at least half again as big as them, and thus they would need years of training for tae kwon do to be much use in that kind of situation.
I saw Akebono in a restaurant in Azabu, Tokyo a few months ago, and he struggled to get in and out of the chair. Even taking into account his yokozuna history, his MMA record is 1-11-0, and so is hardly worthwhile to consider for the OP. If you argue that Akebono is a valid candidate, then the OP’s question is moot.
Gotta say, if a 400-pounder fell on most folk, they’d have a hell of a time getting out from under him!