http://trends.builtwith.com/javascript/jQuery is where we get that statistic from

Wish I could access the full list, but I think that's convincing enough. Thanks for the link.

"powering" is a bit of a weasel word in this case. Plenty of those sites should work just fine without jQuery or any javascript..

So to say jQuery "powers" half the major websites on the Internet is like saying "Cup holders power 99% of all passenger vehicles world-wide".

According to Wappalyzer 60% of sites that use a JavaScript framework use jQuery:

http://wappalyzer.com/categories/javascript-frameworks

It's a different statistic but does make the 50% seem reasonable.

Few of them are major, but the worlds most popular CMS, Joomla, runs 2-3% of all websites, and they will be switching to jQuery (from mootools) later this year.

What data do you have that supports that Joomla is the world's most popular CMS?

It sounds nicely ambiguous. What's a 'major' website?

I would just guess that most major websites are using it.