Still, the family’s progress has a two steps forward, one step back quality. Jim Robinson was born into slavery, but his son, Fraser, ran a lunch truck in Georgetown. In turn, his son, Fraser Jr., struck out for Chicago in search of something better. But he was unable to find work, and left his wife and children for 14 years, according to his son Nomenee Robinson. As a result, Mrs. Obama’s father was on welfare as a boy and started working on a milk truck at 11.

Image President Obama hugged his half-sister, Maya Soetoro-Ng, at her December 2003 wedding to Konrad Ng, third from right, in Hawaii. From left, his daughters, Sasha and Malia; his grandmother Madelyne Dunham, seated; Konrads parents, Joan and Howard Ng, and brother Perry Ng; and Michelle Obama. Credit... Maya Soetoro-Ng

After serving in the Army in World War II and finally securing a job as a postal clerk, Fraser Robinson Jr. rejoined his family. He was so thrifty that he would bring home chemicals to do the family dry cleaning in the bathtub. But his son  Mrs. Obama’s father, Fraser Robinson III  became overwhelmed with debt and dropped out of college after a year. He worked in a city boiler room for the rest of his life, helping to send his four younger siblings to college, then his two children, Mrs. Obama and her brother, to Princeton.

Classroom Values

For all of the vast differences in the Obama and Robinson histories, a few common threads run through. Education is one of them. As a young man, Mr. Obama’s father herded goats, then won a scholarship to study in the Kenyan capital. When Mr. Obama lived in Indonesia as a child, his mother woke him up for at 4 a.m. for English lessons; meanwhile, in Chicago, Mrs. Obama’s mother was bringing home math and reading workbooks so her children would always be a few lessons ahead in school.

Only through education, generations of Robinsons taught their children, would they ever succeed in a racist society, relatives said. “My mother would say, ‘When you acquire knowledge, you acquire something no one could take away from you,’ ” Craig Robinson said.

The families also share a kind of adventurous self-determination. In the standard telling, the Obama side is the one that bent the rules of geography and ethnicity. Yet the first lady’s family, the supposed South Side traditionalists, includes several members who literally or figuratively ventured far from home. Nomenee Robinson was an early participant in the Peace Corps, serving in India for two years; later, he moved to Nigeria, where he met his wife; the couple now live in Chicago. Capers Funnye Jr., a cousin of Mrs. Obama’s and a rabbi, was brought up in the black church, he said, but as a young man, he felt a calling to Judaism he could not ignore.

In daring cross-cultural leaps, no figure quite matches Stanley Ann Dunham Soetoro, Mr. Obama’s mother. As a university student in Honolulu, she hung out at the East-West Center, a cultural exchange organization, meeting two successive husbands there: Barack Obama, an economics student from Kenya, and later, Lolo Soetoro, an Indonesian. Decades later, her daughter Maya Soetoro was picking up fliers at the same East-West Center when she noticed Konrad Ng, a Chinese-Canadian student, now her husband.

Now the Obama-Robinson family’s move to the White House seems like a symbolic end point for the once-firm idea that people of different backgrounds should not date, marry or bear children. In Mr. Obama’s lifetime, racial intermarriage not only became legal everywhere in the United States, but has started to flourish. As many as a quarter of white Americans and nearly half of black Americans belong to a multiracial family, estimates Joshua R. Goldstein of the Max Planck Institute for Demographic Research.