Profits jumped 109 percent last year at the Student Loan Marketing Association, the government-chartered, stockholder-owned corporation that provides a secondary market for student loans.

Sallie Mae, as the corporation is known, said yesterday that it earned $37.8 million ($37.75 a share) during 1982, compared with $18 million ($18.05) in 1981. Fourth-quarter profits for the D.C.-based corporation doubled to $10.8 million ($10.82) from $5.4 million ($5.42). The results set records for Sallie Mae.

At year's end, Sallie Mae's assets, which are insured loans purchased from institutions participating in the Guaranteed Student Loan Program, and warehousing advances, which are loans to lenders for investing in insured student loans, were $6.4 billion, up 33 percent from 1981.

American Realty Trust of Arlington also posted a record year in 1982, earning $5.8 million ($2.61), compared with a loss of $325,789 in 1981. Last year's profits include an extraordinary credit of $1.6 million (73 cents).

Revenues jumped 151 percent at ART from $7 million in 1981 to $17.6 million, a record. Company officials attributed some of the gains to the drop in interest rates.

McCormick & Co. Inc., the Baltimore-based spice maker, reported net income (after peso devaluation charges) of $25 million ($2.02) for 1982, compared with 1981 net income of $29 million ($2.50). Sales increased 9 percent to $717.7 million from $659.8 million.

McCormick previously reported that four devaluations of the peso by the Mexican government in 1982 cut the firm's income by $6.9 million (56 cents), including $2 million (17 cents) in the final quarter.

Net income for the final quarter, after accounting for a peso devaluation, was $13.8 million ($1.12), compared with $17.5 million ($1.41) for the fourth quarter of 1981.