Okay, this is off subject just a bit, but whenever this type of discussion comes up, I find myself bewildered when it comes to the cost of teaching our kids. Here's the thing; where I live, the cost to send my kid to a Catholic school is about $9400 per year. The local public school district, where I send my kids, is currently spending just a bit over $13,000 per year per student.
Now, the Catholic schools have class sizes that average around 20 students per classroom. On the other hand, the local public school district has classes averaging 26 students, and because our school levy failed, they cut teachers and are increasing the average class to 28 students per class.
The teachers in our district have an average salary of $53,000 per year. In the Catholic schools, the average is around $40,000. But here is my problem; the salary is not the issue. Classroom sizes in the public schools are larger, so each teacher is bringing in more money per classroom. I realize there are many other costs besides the teachers. There is maintenance, administrative costs, building costs, and a slewful of other costs. But how is it the Catholic schools can do it for so much less? Where is our money going? It's not the teachers who are sucking the money out of us.
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DISCLAIMER: The below is opinion (unless otherwise noted) and as such I don't plan on going out and doing a lot of research, I do feel however that they are logically derived and will be more than happy if someone provides reputable data that conflicts with these thought. In addition, I'm in no way attempting to imply that the situation is the same for every private v. public school scenario. The thoughts below are in no specific order and do not imply they are sorted by budget impact, they are just my ramblings.

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Transportation:
Ya, you know those big yellow things we get stuck behind from August to June. They are expensive, expensive to buy, expensive to operate, and expensive to maintain. One significant difference that has a pretty big impact on budget is that most public schools provide transportation to/from school while, I believe, most private schools do not. For an school district of size, they will then need to have a whole department dedicated to providing this transpiration. Not only to the buses cost, not only are their fuel costs, but when you own a fleet of buses you have to have maintenance facilities, tools, and repair equipment and a staff of mechanics to maintain them.
Student Population, Just Say No:
By law, public schools are required to service all students in their district. They don't have the option of throwing up their hands and saying "sorry we're full". Like private schools can. Private schools can establish a facility to meet a targeted population and that's it. A student doesn't get in, they don't go to that school. On the other hand public schools are required to maintain in infrastructure able to support the needs of the surrounding community, they don't have the option of saying no. Some school districts probably do this "facilities management" well, other - probably not. What happens when a school gets more students then they can hold, they can't say no, so they have to resort to emergency (read as more expensive) measures which may include rental of another space (with additional transportation costs) or installation of mobile trailers on the grounds which can host hundreds of thousands of dollars (there is the cost of the trailer and then providing utility access).
Student Population, Selection:
The whole idea of "private" school is to not have to service all students that apply. As such private schools have much more flexibility and can admit only the students they want. Maybe it's students of a certain religious persuasion, maybe it's students only of a certain academic caliber, maybe it's excluding students with any type of behavior issues, maybe it's excluding any students that have expensive disabilities. The fact is that public schools are normally required to admit all comers, private schools are not. That means that students with "issues" are normally going to get their education from the public system. Time is money, when the staff doesn't have "issues" to deal with, costs go down. When the system has to deal with "issues" costs go up.
Special Education:
As has been previously mentioned special education is it's own world with requirements mandated by law. The cost for dealing with special needs students can cost the district ten's to hundreds of thousands of dollars for each student. Those additional costs are then averaged over the student body and cause the "cost per student" to be artificially high. Well maybe not artificial, because they are costs - however they are cost that private schools are normally unwilling to bear, unless of course they are a private institution designed to deal with server disabilities, then they often still get to bill the public school for their services. But most private schools are going to look at their bottom line and not take $9400 (to use the figure in the OP) in tuition to then hire a Special Education teacher for a group of 4-6 students or hire a specially trained Student Aid for that student at the cost of $20,000 a year. For the private schools special needs students don't make economic sense.
Homeschooling:
Another area that doesn't get much media attention is homeschooling, and by this I don't mean parents who decide to educate their children at home. I mean students that are so disabled they cannot attend a regular school or those who have such serve disciplinary problems they have been expelled. For medical reasons it may be a long term permanent disability or it may be short term due to illness or surgery. In our district at least, and I think it's pretty common, even though a student is "expelled" the school system may still have a legal responsibility to educate them. That means normally either paying for private school (if one will accept them) or providing tutors in the home.
Personnel Costs:
As has been previously mentioned there are definitely addition school building costs due to base pay and benefit packages. However there are also additional costs based on scale. The costs with supporting a single building can be much leaner then the coast associated with managing an entire school district as an integrated organization. Managing he cost of a single school with a few dozen people and a budget of a couple of million, is very different then the needs of an organization with with thousands (or tens of thousands) and budgets in the hundreds of millions.
Athletics:
This one I did some research on the other night for another posting. In my district 55-60% of the Athletic Program budget (excluding coaching salaries) is paid by the general fund from taxpayer revenue. These items include things like uniforms, equipment, transportation, consumable supplies, security, concession supplies, league referees, etc... 40-45% is covered by athletic revenue, such as ticket sales, concession profits, activity fees, corporate sponsorships/grants etc... 100% of the wages for coaches are paid out of general taxpayer funds. If I remember correctly Title 10 of the United States Code requires public schools to provide equal athletic program opportunities to females if they provide programs for males, regardless of the cost. These athletic costs are part of the school budget and so are reflected in the "Cost Per Student" number as a result. Most private schools that I know of (I'm not saying all) don't provide the same breath of athletic programs that public schools are required to and the cost of those programs are normally covered by: (a) higher activities fees paid by parents, (b) booster clubs, and (c) fund raisers.
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Now in my personal opinion, if in depth analysis were to occur, personnel costs would be the biggest contributor - but each of the items listed above has the potential to impact the cost differential.
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