K12 Inc., the Herndon-based company that is the country’s largest provider of full-time online education, lags behind charter schools and traditional public schools on a broad array of academic measures, according to a new study.

Students enrolled at K12, which provides public virtual education in 29 states and the District of Columbia, lag behind their counterparts on federal and state measures of math and reading proficiency, according to a study released Wednesday by the National Education Policy Center.

The authors, Gary Miron and Jessica Urschel of Western Michigan University, also found that K12 students have lower graduation rates.

Using state and federal data, the researchers examined 2010-11 academic results for K12 schools in five states: Arizona, Arkansas, Idaho, Ohio and Pennsylvania.

They also examined financial reports for those schools for 2008-09, the most recent data publicly available. The five states accounted for more than half of K12 enrollment in 2008-09.

“The results are very weak,” Miron said, referring to K12’s academic record. In states where state education officials graded the quality of schools, 19 of K12’s 36 schools received a clear satisfactory rating, he said.

K12 said in a statement that Miron and Urschel used “incomplete or incorrect school data.”

The company, which began 11 years ago operating in two states, now teaches about 95,000 students.

It operates like a charter school in that it receives a per-
pupil payment from states. But the students work from home, logging on to computers to interact remotely with teachers or do course work.

Virtual schools are championed by some lawmakers, families and school-choice advocates, but have come under increasing scrutiny from critics who say it is an ineffective way to educate students and a bad deal for taxpayers.

Miron and Urschel found that K12 spends less per pupil on average than the states or federal government. “We were kind of shocked by that,” Miron said. “We don’t know where the money is going.”

K12 said in its statement that it is a vendor to public schools and as such “does not ‘spend money’ on anything” but rather supplies a service based on decisions made by public school districts or charter authorities.

In fiscal 2011, K12 had revenue of $522 million — a 36 percent increase from the prior year, according to securities filings.

Its net income after a series of acquisitions was $12.8 million. Ronald J. Packard, the company’s founder and chief executive, earned $5 million in total compensation.

Full-time virtual schools are proliferating. In the past two years, more than a dozen states have passed laws and removed obstacles to encourage virtual schools. About 250,000 students are enrolled in public virtual schools.

A 2009 analysis by the Education Department found that there wasn’t enough research to draw conclusions about how elementary and secondary students fare in full-time virtual schools compared with classrooms.

Meanwhile, a class-action lawsuit filed in federal court against K12 by a shareholder has been amended to include charges from more than a dozen former K12 employees that the company aggressively enrolled students at risk for failure in order to collect public subsidies, among other practices.

A company spokesman declined to comment on pending litigation. But in filings with the Securities and Exchange Commission, the company said it “intends to defend vigorously” against the complaint.