It looks as though Facebook might not be the effective customer engagement engine so many business owners have thought it to be.

On average, only 6% of a brand's Facebook fans actually engage with the brand's Facebook Page via likes, comments, and answers to poll questions (among other ways), according to a new study by Facebook app developer Napkin Labs. The study found over the course of a month, the average Facebook fan will engage with a brand's page less than once.

It turns out, this is usually the company's fault. Riley Gibson, the CEO and co-founder of Napkin Labs, said that most companies simply focus on growing their number of page likes via contests and sweepstakes. But these "like grab" tactics do little to foster lasting, consistent customer engagement.

"It all starts with content," he said. "Like count is becoming less and less relevant. ... Engagement is about creating great content relevant to your customers--content that's helpful and useful to them."

The study also showed that having an engaged Facebook audience is about quality, not quantity. Brands with 900,000 to 1 million fans 60% less engagement than brand pages with 500,000 to 600,000 fans.

Another study conducted by social analytics company Socialbakers suggests that many times, a brand's Facebook fans are trying to engage with the company, but the company is unresponsive. More than 50% of customer questions posted on Facebook went unanswered, according to the study. (An inforgraphic of the study is available below.)