Recently, in Washington, D.C., a five-year-old girl dialed the emergency 911 number to tell police her mother had been shot. The distraught child hung up before giving the address. But police arrived on the scene shortly afterward. The family's telephone number had been automatically displayed on newly installed telephone equipment.
Such quick help was made possible by ''enhanced 911 service.'' This sophisticated emergency phone service pinpoints the number, and, in some cases, the caller's address. Sixteen communities in the United States have enhanced service now, and more are expected to adopt it in the next few years, according to Roger W. Reinke of the Commerce Department's National Telecommunications and Information Administration (NTIA).
''E-911'' service is one example of how emergency phone service is improving. Another is the rapid growth in public-service telephone hot lines - both local numbers and national toll-free ''800'' numbers. These have mushroomed over the last decade, according to American Telephone & Telegraph Company officials. In many communities, the first few pages of the phone book list numbers for immediate help with drug and alcohol problems and various kinds of family crises.
There is little statistical evidence to support claims by public-service organizations that hot lines have often proved as effective in dealing with personal emergencies as 911 has been in responding to crime and fire emergencies. But ''obviously, there is some benefit from them that people wouldn't get otherwise,'' notes Gary Hayes, executive director of the Police Executive Research Forum, a law-enforcement think tank in Washington.
Moveover, spokesmen for organizations such as Parents Anonymous (PA), a national, largely volunteer group devoted to stopping child abuse, cite many examples of children's lives being saved by use of a hot line. PA has a national , 24-hour number, and its chapters in all 52 states have local hot lines.
People who have called a hot line such as PA's before know what to dial and when to call. But the question remains how to help those who are either calling for the first time, or can't recall the number because of a crisis situation. Also, even individuals who have the time and ability to look up a number, or ask ''information'' for one, may be confused about just where to turn. They may end up calling an organization that doesn't meet their specific need.
To help address this problem, Mr. Hayes and other law-enforcement experts, plus a growing number of local telephone company officials, are proposing melding the 911 system with appropriate public-service hot lines. The result, ideally, would be a comprehensive joint effort to assist people in potentially dangerous situations - situations that a 911 operator may not believe worthy of prompt police or fire department aid.
The technology already exists for these efforts to succeed, according to Gene Fredericks, an expert on 911 service for the New York Telephone Company. In fact , adds Chuck Reardon, a spokesman for the New England Telephone Company, the E- 911 system can easily be equipped with a set of buttons to immediately link callers up with public-service agencies.
But E-911 is about twice the cost of regular 911 systems, Mr. Fredericks says. And while some telephone company officials optimistically call E-911 ''the wave of the present,'' severe budget restrictions at both the state and local levels may delay the establishment of more such systems. Hence the further step of linking them up directly to public-service hotlines may be farther away than it appears.
Another factor working against such a link-up is the concern of some law-enforcement officials that people might start calling 911 with problems that are relatively personal and trivial compared with crime and fire emergencies. As Minneapolis Police Chief Anthony Bouza put it, if that happened, ''the essential purity of 911 might be sacrificed.''