Group health insurance premiums rose modestly this year, according to a survey released Tuesday by the Kaiser Family Foundation.

The survey of more than 2,000 employers found that the premium for family coverage rose an average of 4%, increasing to $16,351 this year.

“We are in a prolonged period of moderation in premiums, which creates some breathing room for the private sector to try to reduce costs without cutting back benefits for workers,” Drew Altman, president and CEO of the Kaiser Family Foundation in Washington, said in a statement.

That 4% increase compares with the same increase Kaiser found in 2012 and a 9% increase in 2011. That 9% increase, which was much higher than increases found in several other surveys, may have been an “aberration,” Mr. Altman said at the time.

In addition, a White House official attributed the large health care premium hikes Kaiser found for 2011 at least in part to insurers' miscalculations on the effect of the health care reform law.

• Premium costs are slightly higher at larger employers than at smaller organizations. For example, the average premium cost for family coverage among larger employers, defined as those with at least 200 employees, was $16,715. That compares with an average premium cost of $15,581 among employers with between three and 199 employees.

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• Employees working for smaller companies pay a much greater share of the premium than those working for larger employers. For example, in 2013, employees working at organizations with between three and 199 employees paid an average premium of $5,284 for family coverage, or a little more than $1,000 higher, compared with the $4,226 average premium employees at larger organizations paid.

• Thirty-eight percent of employees with single coverage were enrolled in a plan imposing a deductible of at least $1,000, up from 34% in 2012 and nearly double compared with 2008, when just 18% were enrolled in such a plan.

• The percentage of employers with at least one “grandfathered” health care plan fell to 54% this year, down from 58% in 2012 and 72% in 2011. Under the health care reform law, grandfathered plans are exempt from certain requirements, such as offering full coverage for preventive services. But to qualify for grandfathered status, employers face limitations on how much they can shift funding cost increases to employees through boosts in deductibles and copayments.