L.A. County’s jobs picture showed broad improvement last month as the unemployment rate dipped to 12.2 percent and the county added 32,000 payroll jobs over the month, according to state figures released Friday.

Led by gains in government and private education, the unemployment rate dropped last month from 12.4 percent in September. The rate is down substantially from 12.9 percent recorded a year earlier, according to household survey figures from the state Employment Development Department.

However, the unemployment rate still is above the statewide average of 11.7 percent and is well above the 9 percent national average. And the county’s two largest cities, Los Angeles and Long Beach, still have unemployment rates of 13.1 percent, nearly a full percentage point above the countywide average.

A separate survey of employer payrolls showed a broad jobs recovery taking root in the county. Nine of the 12 industry sectors tracked by the state showed job gains over the month, led by state and local government (up 13,500 jobs) and private education (up 7,100). Even the long slumbering construction industry showed life, gaining 2,000 jobs over the month. The only sector that lost jobs over the month was information, which dropped by 1,300 jobs.

The gains were more modest in the year-over-year payroll jobs figures. Between October 2010 and October 2011, the county gained 19,000 jobs, an increase of 0.5 percent. The largest increases came in private education, professional and business services, health care and the film and television industry. Local government and financial activities posted drops.

Adjusting for seasonal hiring factors, county employer payrolls showed a gain of 1,600 jobs this month, according to figures from Los Angeles-based Beacon Economics.

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