What's the difference in saying "the average family" vs. "the family average"?

Consider these two sentences:

A Scranton family has an average of 2.1 children.
The average Scranton family has 2.1 children.

I've seen both used. The first one seems reasonable, but the second one seems wrong because you're applying average to family, not to numbers.

To me, saying "the Jones are an average family" seems legitimate, but if the Jones have 2.1 children, they're anything but average.

Is the second sentence grammatically correct? Is it understood to mean the same as the first sentence?

Inspired by the question, Meaning of the expression "2.1 kids".

As you pointed out, the two sentences would typically be interpreted as having the same meaning.

A Scranton family has an average of 2.1 children.
The average Scranton family has 2.1 children.

The prototypical (average) family has 2.1 children. "The average family" in this case is a non-existent family, one whose properties are all equal to the statistical mean of the implied domain. This is a construct used for some statistical analysis.

The Jones are an average family.

The use of "average family" in this case means that this specific family is not an extraordinary case. Their properties are probably close to the statistical mean. However this is very much dependent on context. Perhaps the Jones are average with respect to income but not necessarily number of children. Using this phrase would not indicate that the Jones family has exactly 2.1 children.

In the first sentence, average refers to the the result obtained by adding together all the children in Scranton, and dividing this total with the number of families in Scranton.
In the second sentence, average family means an ordinary family.

Both sentences are grammatically correct.