The number that now divides the parties was introduced by Mr. Obama in 2007, in the early days of the presidential campaign, when he promised to extend the Bush tax cuts for families that made less than that amount. “I can make a firm pledge,” Mr. Obama said in September 2008. “Under my plan, no family making less than $250,000 a year will see any form of tax increase. Not your income tax, not your payroll tax, not your capital gains taxes, not any of your taxes.”

When policy makers talk about income, however, they do not mean the amount of money a family earns; they mean the portion subject to taxes. For those who itemize their tax deductions, the government does not tax interest payments on mortgages or charitable donations, among other things, up to certain limits. As a result, two families with the same incomes will most likely have different taxable incomes.

To guarantee that tax rates do not increase for any family making less than $250,000, the Obama administration proposed in 2009 to raise marginal rates on taxable income above roughly $230,000 — because the minimum amount of income a family is entitled to shelter from taxation is roughly $20,000.

But the average amount families in that income range are entitled to shelter from taxation is much larger, closer to $60,000. In other words, families with taxable income of $230,000 on average earned about $290,000 in 2009.

“They wanted to be able to say that ‘Absolutely nobody making less than $250,000 could possibly pay higher taxes under our plan,’ ” said Robert S. McIntyre, the director of Citizens for Tax Justice, a liberal advocacy group. “So they had to assume the most ridiculous assumptions, that even if you’re a childless couple with no itemized deductions making $250,001, your taxes still won’t go up. They figured that if this couple existed and their taxes went up, somebody would find them and jump on ’em.”

Furthermore, to remain consistent with the president’s original promise, the administration has adjusted the original numbers for inflation. When Mr. Obama says $250,000, the White House says he means “in 2009 dollars.” It is now proposing to raise marginal rates on families with taxable incomes above $246,000 — meaning, on average, families earning more than about $305,000.

While the president has said that he wants to raise tax rates for the top 2 percent, only about 1 percent of taxpayers will face higher marginal rates, according to an analysis by the Tax Policy Center, a widely respected research group.