WASHINGTON — The Supreme Court led by Chief Justice John G. Roberts Jr., the conventional wisdom goes, is exceptionally supportive of free speech. Leading scholars and practitioners have called the Roberts court the most pro-First Amendment court in American history.

A recent study challenges that conclusion. It says that a comprehensive look at data from 1953 to 2011 tells a different story, one showing that the court is hearing fewer First Amendment cases and is ruling in favor of free speech at a lower rate than any of the courts led by the three previous chief justices.

The study arrives as the Supreme Court prepares to consider two major First Amendment cases. On Tuesday, the court will hear arguments in Federal Communications Commission v. Fox Television Stations, No. 10-1293, which asks whether the First Amendment allows the government to regulate vulgarity in broadcast programming. Next month, the court will consider United States v. Alvarez, No. 11-210, which asks whether the government can make it a crime to lie about receiving military decorations.

In neither case is a ruling in favor of the free speech argument assured. Indeed, how the court decides the cases will help determine whether the court’s reputation as a fierce protector of the First Amendment is deserved.