Are there some numbers on how much vertical distance can people make in a day’s hike? I know that greatly depends on fitness and other factors, but I’d be happy with some statistical curve that would say something like 80 % of people should make 1000 meters up and so on.
Actually, it depends on pretty much everything: Weather, terrain, fitness, group size, and of course, how long a day is.
It's better to think of 'How much can I do in an hour, and how many hours do I want to walk/climb for?'.
I'd start using Naismith's Rule. But, as you get experience about your own group's speed and fitness, adjust it up or down as appropriate.
Here's one data point, also based on the Grand Canyon.
I'm a 40 year old male, of average fitness and slightly overweight (5'10", 215 lbs.), and ascending the Grand Canyon (4320', ~1300 m) took me almost exactly one day. By contrast, the descent took me 3 hours. (The rule of thumb at the Grand Canyon is that every hour down takes two hours up.)
Later that same trip, an elevation change of 2100' (from the base of Zion Canyon to past Observation Point, ~640 m), took me a little less than 4 hours - from 4:30 in the afternoon until just before sunset around 8. In my 20s, I trekked from Pokhara to Jomsom (in Nepal), and ascended Poon Hill (elevation gain = approx 1200m). The climb up took a day, the climb back down, about half a day. (The stairs at Ulleri are killer!) The elevation to Tukche is then about the same, but the straight line distance means that it takes another two days.
In my 30s, it took about 4 or 5 hours to climb to the top of Yosemite Falls, approximately 3000' (~900 m) to the top. It then took about 2 hours to climb back down.
I give these data points not as an average, but to illustrate a few points:
Figure how much this guy covers in a day:
Here's one I didn't know about until today:
Since people's capabilities vary enormously, I'm going to answer in terms of AMS - altitude sickness. Although susceptibility to AMS also varies, there are some general rules of thumb for how much a person "should" climb in a given day which are worth knowing.
For starters, at up to about 3000 meters, you don't have to worry about it too much. What I mean by this is that theoretically, you could ascend from 0 to 3000 in a day and not be at excessive risk. An example of this would be hiking to the bottom of the Grand Canyon and back up - 1700 meters each way - which, if you're in great shape, is possible without risking AMS (but crazy for all the regular physical reasons).
At higher altitudes you pretty much have to go gradually to let your body acclimate. If you're planning a trek to the Everest Base Camp (5380 meters) in Nepal, a rate of 300 meters per day is generally suggested. Less than that means that though you might be fine for a while, eventually the change will catch up to you and then you'll need to stop until you're better.
I know that some people climb Mount Whitney (4421 meters, in the US) in a single day, but I wouldn't recommend it - at 4000 meters changing altitude rapidly is very likely to make you sick. They manage to do it because they both ascend to the summit and descend in a single day (AMS symptoms usually begin after a few hours delay) and therefore are back at lower altitudes before the symptoms kick in.
I'll close with some personal experiences. On the way to the Everest Base Camp, I decided to "skip" a day and walk to segments back to back- 600 meters change, at around 4200 meters altitude- and I became sick that evening, although my companion was fine. In a later trek (a few years later) my companions and I averaged about 500 meters/day to a maximum altitude of 4000, and all of us were fine.
Like many other things in the "Great Outdoors", it's best to plan conservatively until you feel comfortable with the subject and have more experience.
In good terrain, I use the thumb-rule: 10 minutes per km of distance + 10 minutes per 100m of altitude. This works on easy trails in moderate conditions (no heat, no high altitude, etc.), and often translates to roughly 400m/hour. This does not include breaks; at this pace, I need a ~15 minute break every two hours. I've never done more than 2000m in a day, though.