How long has the headmaster/principal been in office?
This question speaks to the stability of the school. If the headmaster or headmistress (also called head and occasionally director) has been there for a couple of years, that's a good sign. Private school heads will stay forever if they are doing a good job and the trustees are satisfied with his job performance. Nowadays a private school head is the de facto CEO of the school. But his major responsibility is going to be in the area of fundraising. Public relations is another part of his brief.
If the door to the headmaster's office has become a revolving one with several heads coming and going over a period of a few years, you might want to find out why they didn't stay. Most private schools conduct national searches for a head of school and involve the school community in the process. So it would be unusual for a school to get the fit wrong.
How large is the endowment?
The size of a private school's endowment is another measure of the school's financial stability. Think of an endowment as the school's savings account. The endowment acts as a cushion in tough times. If the headmaster and fundraising staff are doing their jobs properly, the endowment should be growing steadily as grateful alumni and friends of the school remember the school with bequests and special gifts. Endowments also contribute significantly to the financial aid programs of most schools. Financial aid awards allow students who could not otherwise afford to attend receive a private school education.
The endowment will be small or non-existent in a new school. There's nothing wrong with that. Just be prepared to contribute to help get the school on the proper financial footing. On the other hand if the school has been around for decades and has burned through its endowment, that is probably not a good thing. Prudent management of school resources is always a good thing.
Financial condition of school
You can make your own judgement about a private school's financial condition by reading its Form 990. All private schools which are non-profit entities are required to file a Form 990 which is their tax return. These Forms 990 can be found online in several places on the web including The Foundation Center and Charity Navigator.
If the school is living off tuition income, that will be apparent in those financial documents. An endowment gives a school another source of income, meaning that tuition doesn't have to cover all expenses.
Most private schools have a skilled financial professional running operations on the back end. The federal, state and local reporting requirements in addition to the financial management duties require experienced, capable people to run the school efficiently.
Number of faculty with advanced degrees
Most private schools pride themselves on attracting and retaining well-qualified teachers. As you peruse the faculty listings, you will inevitably feel the pride as the school describes the degrees held and so on. You expect your child to be taught by somebody who knows his subject. The schools understand that and work very hard to attract and retain faculty who will measure up to your expectations. You can expect to see 70-80% of faculty with advanced degrees. Often those degrees will be in their subject.
As with the headmasters, private school teachers tend to stay forever if they like what they are doing and are reasonably well-looked after. So, if you see teachers coming and going, find out why that has happened. An unhappy teaching staff is not a good sign.
Choosing a school is a lot of work. You must do your due diligence, ask questions and gather the information which you need to make an informed decision. A lot is at stake besides the huge expense. Your child's happiness and well-being are of paramount importance.