Is there any attempt to measure who is using the Stack sites or how that audience has changed over time?
What does the audience look like today? What are you trying to make it look like in 6 months? In a year? In 2 years?
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All Stack Exchange sites have a site-analytics privilege. It is awarded at...
This provides basic data, on the posts and traffic that a site may get.
There is also the Stack Exchange Data Explorer which is available to everyone, and free to use. There are many queries that have already been written and are free to use.
This is data that you can collect and analyze yourself.
What are you trying to make it look like in 6 months? In a year? In 2 years?
Well, some sites have goals, and others don't. The goal of Stack Exchange is not to become the smallest community on the internet within 10 years, but to create a compilation of knowledge that anyone interested can participate in and learn more about. This is how SE thrives now, and will continue to thrive in the future.
As rene noted, there are also institutions that make studies out of data that SE provides. You can learn more from this blogpost.
Some (necessarily approximate) demographic data is available from Quantcast: follow the link
quantcast.com/site_domain, for example https://www.quantcast.com/stackoverflow.com#!demo
According to Quantcast:
and so on...
The gender distribution is different, e.g., for Seasoned Advice: https://www.quantcast.com/cooking.stackexchange.com#!demo