''I'm angry at myself for having been put in the position - put myself in the position - of having to make this choice,'' Mr. Biden said in a firm, clear voice. ''And I am no less frustrated for the environment of Presidential politics that makes it so difficult to let the American people measure the whole Joe Biden and not just misstatements I have made.'' 'Shadow' of Mistakes

Mr. Biden said that ''the exaggerated shadow'' of his mistakes had ''begun to obscure the essence of my candidacy and the essence of Joe Biden.''

Reflecting the immediate priority of the remaining Presidential contenders in winning over some of Mr. Biden's financial backers, Ann F. Lewis, a Democratic consultant, said that over the coming weeks, ''members of the Biden finance committee are going to be the most popular people in town.''

Mr. Biden stood at about 10 percent in public opinion polls in Iowa, where the first key nomination test will take place. According to a survey by The Des Moines Register, Mr. Biden's supporters were likely to disperse widely among the remaining Democratic Presidential candidates.

When Mr. Biden's supporters were allocated to the remaining candidates on the basis of their second choices, Representative Richard A. Gephardt of Missouri led in Iowa with 21 percent, followed by Mr. Dukakis with 16 percent, Senator Paul Simon of Illinois with 13 percent and the Rev. Jesse Jackson with 11 percent. Bruce Babbitt, former Governor of Arizona, had 9 percent, Representative Patricia Schroeder of Colorado had 6 percent and Senator Albert Gore Jr. of Tennessee had 2 percent.

The Register's findings were based on a re-analysis of a survey of 316 likely Democratic caucus participants conducted in the third and fourth weeks of August. Joins Hart as Casualty

Mr. Biden was the second candidate, after Mr. Hart, to be knocked out of the contest by a quick succession of revelations in the press. And he was among the first to suffer from the spread of video-technology, which can make even the most intimate living-room campaign gathering into a national political event.