The Washington Post High School Challenge is designed to recognize schools that challenge average students. These 22 top-performing schools, listed in alphabetical order, were excluded from the list because, despite their exceptional quality, their admission rules and standardized test scores indicate they have few or no average students. Non-neighborhood schools with SAT or ACT averages above the highest averages for neighborhood schools nationally are put on this list:

Accelerated Elementary and Secondary School, Tucson, Ariz.

This tiny charter school in 2011 had only 4 graduating seniors, whose average SAT score was 2115.

Bergen County Academies, Hackensack, N.J.

A collection of seven career-focused academies where students have an extended school day.

Biotechnology High School, Freehold, N.J.

A demanding program started in 2005, with two large research rooms, four state-of-the-art science labs and most of the senior class earning International Baccalaureate diplomas.

Bronx High School of Science, New York

One of the most famous schools in America. It has a richly talented, ethnically diverse student body.

Davidson Academy of Nevada, Reno

The tiny school — just six graduating seniors in 2010 — opened in 2006 on the campus of the University of Nevada at Reno as a “university school for profoundly gifted students” under a state law.

Gatton Academy of Mathematics and Science, Bowling Green, Ky.

Juniors and seniors from across the state are selected by scores, grades and essays to live in their own Western Kentucky University residence hall, earning college credit as well as completing high school.

High Technology High School, Lincroft, N.J.

The school, opened in 1991, uses the increasingly popular High Tech High name for magnets emphasizing hands-on learning. It is run by Monmouth County and Brookdale Community College.

Hunter College High School, New York

One of the city’s legendary public high schools, with a program for seventh- through 12th-graders administered by Hunter College. It was an all-girls school until 1972.

Illinois Mathematics and Science Academy, Aurora

“Wayne’s World,” the Mike Myers Saturday Night Live sketch and film, is not Aurora’s only claim to fame. IMSA is a state-funded boarding school. It takes 10th- through 12th-graders and has a strong mentoring program.

International Academy, Bloomfield Hills, Mich.

The magnet school for 20 Oakland County school districts has three campuses. All students take the International Baccalaureate diploma program.

International Community School, Kirkland, Wash.

Students are selected through a lottery to attend this school focusing on international awareness. It is one of the few elite public schools without a selective admissions system. Instead, as happens sometimes, the lottery participants self-select into an academic powerhouse.

Louisiana School for Math, Science & the Arts, Natchitoches

Sophomores, juniors and seniors who survive the tough admissions process live at the boarding school established by the state legislature in 1982.

Maggie L. Walker Governor’s School for Government and International Studies, Richmond

Unlike the science-math orientation of most of the public elites, the focus of this school is on world cultures and building students’ leadership skills.

North Carolina School of Science and Mathematics, Durham

This school, established in 1980 in an abandoned hospital, started the small but interesting trend of state-created boarding schools drawing bright and ambitious high-schoolers from across the state.

Northside College Prep, Chicago

Opened in 1999, the selective magnet school was the first new high school in Chicago in 20 years. It has 1,100 students whose academic classes are at the honors and AP level only.

Oklahoma School of Science and Mathematics, Oklahoma City

A state-funded boarding school that teaches all courses at the university level.

South Carolina Governor’s School for Science and Mathematics, Hartsville

Another state boarding school, this one for 11th- and 12th-graders across the state.

Stuyvesant High School, New York

It has been teaching the city’s most academically ambitious students for several generations. It offers about 55 AP courses every semester and has plenty of courses above that level.

Thomas Jefferson High School for Science and Technology, Fairfax County

It is the most selective public high school in America and draws mostly from the affluent households of Northern Virginia.

Union County Magnet High School, Scotch Plains, N.J.

This selective-admission school also focuses on science, math and technology.

University Laboratory High School, Urbana, Ill.

There is competitive admission for this day school on the campus of the University of Illinois. It makes good use of its higher-education environment.

Whitney High School, Cerritos, Calif.

Like Jefferson, this suburban version of the New York City super-schools has a very competitive admissions system. Unlike students at the state boarding schools, those at Whitney go home at night.