Even when tax revenue is scarce, lawmakers rarely ask a key question before they vote on education budgets: Which public schools produce the most bang for the buck?

A new analysis of spending patterns and test scores in school systems nationwide found a surprising result: Manassas Park and Calvert County have the most efficient schools in the Washington area.

The Center for American Progress, a liberal think tank, also found that Fairfax County schools are reasonably efficient compared to others in Virginia. Montgomery County’s stack up well in Maryland.

Efficiency ratings, of course, are debatable. There are long-running disputes about the reliability of test scores and how much money schools need to help poor children overcome academic disadvantages. It is especially hard to compare Manassas Park, which has about 3,000 students, to mammoth Fairfax, which has more than 170,000.

But debate is exactly what the report’s authors want.

“At its core, you want to compare outcomes to spending,” said Ulrich Boser, the lead analyst on the project. “Let’s start a conversation. It’s just very important in this time of sagging revenue.”

The Obama administration has picked up on the theme as the flow of federal aid from the 2009 stimulus law has dried up. Education Secretary Arne Duncan in November declared that the “new normal” for schools will be doing more with less.

“It’s time to stop treating the problem of educational productivity as a grinding, eat-your-broccoli exercise,” Duncan said. “It’s time to start treating it as an opportunity for innovation and accelerating progress.”

Boser said Florida and Texas are the only states that rate schools formally on productivity.

Across the country, school officials this year are wrestling with how to cut costs but save essentials. Many are freezing salaries, raising class size and pondering teacher layoffs. Others are cutting central office staff, bus service, counselors, summer school, arts classes and extracurricular activities. These cuts are provoking outrage in many places because there is little consensus on what are educational necessities and what are frills.

For its study, the center analyzed spending and achievement data from the 2007-08 school year for about 9,000 school systems. Proficiency rates on state tests were examined for fourth and eighth grades and high school.

Researchers compared districts to peers within their states, omitting the District of Columbia and a handful of states from the analysis. The analysis took into account cost of living and how many students in a given district come from poor families, have disabilities or are learning English as a second language.

The study found efficiency varies widely within states. In New York, the range of spending among top-achieving districts was more than $7,000 per student. In California, it was almost $8,000.

In the Maryland suburbs near Washington, Calvert County schools had the top rating even when results were sliced three different ways.

Per-pupil spending in the 17,000-student system on the western shore of the Chesapeake Bay was slightly more than $8,000 when adjusted for cost of living and other factors, about $1,000 less than in Anne Arundel County and $2,000 less than in Montgomery. Calvert also had high test scores.

Calvert Superintendent Jack Smith said the county has a lean central office and requires nearly all teachers to have full classroom duties. In other systems, Smith said, schools often employ numerous teaching specialists and other personnel who are not assigned to particular classes.

“We don’t have a lot of other people doing a lot of other work outside the classroom,” Smith said. “Our only purpose here is for students to learn. That helps us make decisions about how to spend our money.”

Elsewhere in Maryland, on an index called “basic return on investment” — one of three methods the center used for comparison — Frederick County received a top mark. Schools in Montgomery, St. Mary’s and Anne Arundel counties were rated just behind. Then came Howard County, with high achievement at a relatively high cost. Prince George’s and Charles counties had the lowest efficiency ratings among Maryland’s suburbs.

Montgomery officials said their ratings were among the best for the nation’s largest school systems.

In Northern Virginia, Manassas Park and Prince William County got the highest marks on the basic index. They were followed by Fairfax and Loudoun counties and Falls Church. School systems deemed less efficient, in descending order, were Manassas, Arlington County and Alexandria.

Loudoun schools spokesman Wayde Byard said the study failed to account for explosive growth in the county, which now has more than 63,000 students, and significant budget cuts. “This is a snapshot of where we were three years ago,” he said. “So much has changed since then; it no longer tells our story.”

Alexandria’s low rating reflected its relatively high spending — more than $10,000 per pupil, according to the study — and what Superintendent Morton Sherman called “achievement issues which are long-standing.” Neighboring school systems had higher achievement while spending thousands of dollars less per pupil.

Sherman, who took over after the school year that was analyzed, said the city’s schools are moving in the right direction. “We have a great deal of work to do,” he said. “Early achievement data shows improvement across the board.”