After years spent measuring and analyzing the problem, the city is now asking organizations to work on fixing it. In recent months, 33 cultural institutions on city-owned property submitted plans to boost diversity and inclusion among their staff and visitors; if they failed to do so, the city warned, their funding could be cut.

The plans were filled with organizational charts and multistep processes for diversifying their employees and making people of all backgrounds feel comfortable visiting their sites.

“This is what our city looks like, and this is what we should look like,” said Shanta Thake, the senior director of artistic programs at the Public Theater, which set a goal for its full-time staff to be no more than 50 percent white by 2023. (It’s currently 57 percent white.)

But the question of accountability remained: How far would the city go to hold these groups to their plans?

The demographic survey, which was completed by Southern Methodist University and largely funded by a grant from Deutsche Bank , collected information from employees and volunteers at the 33 institutions on city-owned property and 32 others that receive city money . But while the study showed, for example, that 11 percent of arts workers surveyed are Hispanic, compared with 29 percent of New York’s population, and that 10 percent are black (compared with 22 percent), the city does not know which organizations have the biggest diversity problems. The study was based on employees’ self-reporting, and the institutions themselves were not required to submit their comprehensive data to the city.