Given a map layer of point locations, and a second map layer of census tract polygons for the same area, how would you create a map layer that shows the average walking distance to the closest point for each census tract?
Euclidean Distance and Zonal Statistics (As Table) will tell you the average distance to the closest point for each polygon.
It's worth considering that Euclidean distance is not walking distance. Walking routes are restricted: a person can't (reasonably) walk over a building, an interstate highway, a river/stream, etc.
A more accurate method would be to use Network Analyst, but you would need to have a sidewalk/path shapefile for the area.
It is also possible to define "blocked" areas (buildings, waterways, etc.) and run an analysis with the Cost Distance tools, which are basically Euclidean Distance plus additional considerations -- but again, you'd need additional data.
That being said, Euclidean Distance can still be a useful approximation, particularly if you're interested in relative proximity to the point locations -- in other words, that Tract A is (on average) twice as far as Tract B.