The average American household's after-tax income rose sharply in 1983 to $20,001, with a major assist from President Reagan's 1981 income tax cut, the Census Bureau reported yesterday.

A bureau study showed that, contrary to a popular impression, the elderly are not especially poor and have higher per-capita income than most other age groups. It also showed that only 7.6 percent of households below the government-set poverty line in 1983 paid federal income tax, but about two-fifths of them paid Social Security or property taxes.

Yesterday's report is the fourth in a series of studies attempting to determine how much of their cash incomes American households have left after deducting payments for four major taxes: federal income tax, state income tax, Social Security payroll taxes and property taxes.

The study found that, in 1983, the nation's 85 million households averaged $25,401 in gross income, paid an average of $5,400 in taxes (21 percent), and ended up with after-tax income of $20,001. A household may be one or more persons.

The bureau found that, compared with 1982, income before taxes -- in constant dollars after accounting for inflation -- rose 1.2 percent. But after-tax income rose 2.4 percent, showing that tax changes had a clear effect.

The bureau said the major reason that after-tax income rose more was the 1981 tax cut, which reduced federal income-tax rates about 10 percent. People didn't pocket all the savings because state income taxes and Social Security taxes went up, but there was enough left to leave them with higher after-tax income per household.

The overall 21 percent that people paid for the four different types of taxes included in the study compared with 22 percent in 1982 and 23 percent in 1981.

The study found that nine-tenths of all households paid at least one of the four taxes: three-quarters paid Social Security taxes, three-quarters paid federal income taxes, three-fifths paid state income taxes and three-fifths paid property taxes on their homes.

For the population below the poverty line, however, the figures were quite different. They averaged $4,484 in income before taxes, paid $237 each in taxes and ended up with $4,247 per household. Moreover, though nearly half paid Social Security taxes and a third paid property taxes, only 7.6 percent paid federal income taxes and about twice that percentage paid state income taxes.

The study showed that on a per-capita basis, which takes into account the number of people in different types of homes, households headed by persons of age 65 or over had a per person income after taxes of $8,113. This is higher than any age group up to age 50 but a bit lower than the age groups 50-54 ($8,614), 55-59 ($8,996) and 60-64 ($8,961).

The study showed that white households, with $20,751 after-tax income, were financially far better off than the households of blacks ($13,673) or Hispanics ($16,076). Among family groupings, households consisting of women with children but with no husband were at the bottom of the range: $11,302, less than half the figure for married couples.

The West had the highest average after-tax income, $21,153, the South the lowest, $19,474. The Northeast showed the biggest gain over 1982 in constant dollars, 3.9 percent, the South and West each rose 2.5 percent but the Midwest only 0.9 percent.