I really like the idea of Stack Overflow, but I wonder if anyone knows (maybe the site admins, if they're reading) how many new questions are being asked each day.
Is it around 5,000?
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It depends on the day of the week. Recently it's around 10,000 from Monday to Friday and around 4,500 on weekends:
You can find current values on https://sostats.github.io - it's a website presenting the latest statistics about Stack Overflow (questions, answers, comments and users). It's fed with data retrieved from the Stack Exchange API
Yet another way of retrieving this figure (also for other sites than Stack Overflow) is a SEDE query. Note that the data is refreshed only once a week, on Sunday, so it does not give real-time results. The query could be extended to include only questions with a certain tag, if you like.
The API stats route has an entry that shows how many questions a site gets every minute:
So, around 4421/day as of this post
Per the page https://stackexchange.com/sites#questions in January 2018 Stack Overflow receives about 6,900 new questions per day, about 5 per minute.