While it is widely believed that the hole handicap is a measure of a holes difficulty this is not strictly correct according to the USGA.
The USGA states “The stroke hole allocation (hole handicap) should not be based on the difficulty of a hole, but rather where players with higher handicaps need strokes in order to obtain a half in match play. ”
So even with this explanation from the USGA most players would agree that often the #1 handicap hole on a course is perceived to be one of the easiest holes and the #18 handicap hole is often times one of the easiest holes on the course to par.
In order to see just how right or wrong this belief that hole handicap = difficulty I have taken the scores from roughly 27,000 scores logged on Golfingstat.com and graphed the average (mean) score relative to par (e.g. birdie = -1, par = 0, etc.) against the hole handicap number.
Here is the table to support that graph:
As you can see in the table and graph the holes with the lowest 4 handicap numbers (#1 through #4) are significantly more difficult than the average hole. Also the holes with the highest 4 handicap numbers (#15 through #18) are significantly easier than the average hole. So just looking at these 8 holes it seems that the belief that the hole handicap number is a good measure of the hole difficulty holds up.
The middle 10 handicap holes do not really seem to be as closely mapped to their actual difficulty as they have some weird anomalies such as #11 being on average significantly more difficult than #12.