When Alexa Aviles received her census form in 2010, she was frustrated by the choices. Like all Hispanics, Ms. Aviles, a Puerto Rican who lives in Brooklyn, was first asked to identify her ethnicity and then to answer a question about her race. Ms. Aviles, 41, who works for a nonprofit, thought, “I’m all of these!” In annoyance, she checked Hispanic, and then identified herself as white, black and “some other race.”

Mustafa Asmar, a Palestinian-American waiter in Paterson, N.J., does not like his options either. Arab-Americans are broadly classified as white in the census. “When you fill out white or other, it doesn’t really represent the Middle Eastern population,” said Mr. Asmar, 25. “I don’t feel like I’m white. I don’t know what else to put.”

As the United States becomes more diverse, the Census Bureau is grappling with how to accurately classify race and ethnicity in its next decennial count in 2020. It is an issue that plays out in divergent ways for different groups. Many Hispanics, like Ms. Aviles, are frustrated that they are prompted to select from racial categories that they believe do not represent their identity.

Many Arabs have the opposite concern: They are not asked a separate ethnicity question and are typically categorized as white, a label that many feel does not apply.