We'll have a look at how gun fatalities correlate to variety of social phenomena, and I'll just let the cat out of the bag early by saying that the one thing that it has absolutely no correlation with is the incidence of poor mental health. Let's work our way down.
For these charts I've plotted the numbers for a variety of social phenomena on the X-axis vs gun deaths on the Y-axis. Each point represents the paired data for a single state. In each graph, the best 10 (in green) and worst 10 (in red) States for gun deaths are indicated by an oversized dot. The best of best and worst of worst States are in the biggest square dots.
Here are the worst States, ranked by Gun death incidence, 2007 data.
New Mexico 15
And the best.
Rhode Island 3.4
New York 5
New Jersey 5.2
New Hampshire 5.5
South Dakota 6.1
The strongest correlation among all the things I looked at was with Vehicle Deaths, where there is a strong correlation with an R of .8, and R^2 of .63.
The best way I can rationalize this is by grasping for a common cause - perhaps substance abuse plays a part. Note the tendency of best states to cluster at the bottom left and worst states to cluster at the top right. What we will see quite frequently in these graphs is the best States will cluster more tightly, and the worst states will spread out more, right to left.
The chart for the Total Death Rate looks pretty similar. I guess it makes some sense for these two to move together, though they differ in quantity by about two orders of magnitude. At R = .7 and R^2 = .49, there is reasonably strong correlation.
Suicide rate is where I expected the strongest correlation. Alas, twas not to be. Still, with R^2 at .42 (indicating a correlation coefficient at a quite respectable .65) the correlation is significant.
Now, this one is a surprise. There is identical correlation (R = .65; R^2 = .42) to the infant mortality rate.
Correlating gun deaths to the incidence of violent crime seems intuitively right. But R is only .5, with R^2 at .26, so this is bordering on weak correlation.
You get the same correlation to the instance of low birth weight.
What about other health-related events?
Not particularly strong correlations, but something, anyway.
This is what a total lack of correlation looks like: an essentially horizontal trend line (i.e. no trend at all) and correlation near ZERO.
I looked at a number of things that would not be expected to correlate with gun deaths, and found various levels of correlation, from strong to weak. What these things may indicate is up for speculation. Thom Hartman has mentioned studies that correlated all manner of social ills to wealth disparity, and that is certainly one sitation that has gotten much worse in this country since 1980.
It's pretty clear that Louisianna, Mississippi, and Alabama are lousy places to live, but great places to die.
The only social variable in this array that has absolutely no correlation with gun deaths is poor mental health.