What is the average deviation from the PAPI glide slope at the threshold?
If one approaches with three red and one white light what will be the
  average deviation from the standard threshold height of 50 feet?


Short answer

It depends on the location of the PAPI (and its slope). For a 3° PAPI at 1,000 ft (305 m) from the threshold, three red lights means you cross the threshold between 43 and 49 ft.


PAPI heights at runway threshold



Explanation

There are different ways to calculate this. One is to start from the distance of the change from  red to white for the third light, which is given in this table (line B):


Source

The right side is for APAPI, let's ignore it. d2 is the distance from the PAPI (not from the threshold) an aircraft flying at a height of 1,000 ft sees the system switching from 3 red lights to 2 red lights.

If at this point the aircraft starts following the B glideslope, it will maintain this 2/3 red lights limit until landed. It will have lost 1,000 ft over a distance of d2.

A simple application of Thales theorem, about proportionality, says:

height at threshold = 1,000 * d / d2


where d = distance PAPI to threshold.

Example:


PAPI slope 3°
d2 = 3.33 NM = 6,167 m
Distance PAPI to threshold = 305 m


so height at threshold = 1,000 * 305 / 6,167 = 49 ft

If we do the same calculation for the other light transitions, we get: 60, 55, 49, 43 ft respectively.

Actual height

To elaborate on CVn's comment: A PAPI isn't corrected for parallax error. The sensor being the eyes, and having a difference between eyes height and wheels height, the PAPI glide path is always higher than the landing wheels.

In particular when the glide path crosses the threshold at 50 ft, the wheels are a bit lower, the actual height depending on the particular aircraft configuration and attitude. This error comes in addition of the glide path inaccuracy itself (+/- 3 ft at the threshold between lights + angular setup error). A PAPI is not a precision instrument, and the landing has to be done visually.

Larger aircraft performing precision landings are fitted with radar-altimeters located on the belly, well below the pilot eyes and close to the main wheel. Their raw value is usually corrected, by adjusting the aircraft installation delay beyond what is required for line length cancellation, to indicate the main wheels height.

This improved precision allows for lower decision heights (the point where the pilot is not allowed to pursue the landing if the runway isn't in sight).