Accepted Solutions
I don't know whether to suggest making the selected records the default or not.
I am torn between having bogus stats when a few are selected or all the stats when you don't want them.
A flashing icon perhaps 
I have a selection in the attribute fields, but Filter by selection isn't an option for me. Thoughts?
Also, is there a way to not create a distribution chart every time I want to run statistics?
Once you have your selection, you can use ctrl + shift + D to open the data engineering window. On the left side of the pane click and drag the field that you are interested in into the middle and then click the Calculate button. (In my case, I'm interested in the SUM of the length fields). Make sure to click the calculate button every time you make a new selection.
If you used the Attribute Editor and click on the selected part (in the red box in the lower right) it should have shown the selected features when using the Statistics on your field. It will bring up the Char Properties pane and then you can see the Selection listed there beside the entire dataset. This is of course if you were doing it this way and not the data engineering method as it looked like you originally did since you had the distribution chart showing (which I don't use or need either)...
Sharing for posterity. I had this same question except I wanted the sum of the selected records, not the mean, so the solution given above didn't help. Turns out all I needed to do was widen the Chart Properties pane. Statistics for the selection are to the right of the statistics for the dataset.
