This is a visualisation of the cross-sectional profiles of 1549 aerofoils found in the UIUC Airfoil Coordinates Database, all relating to real lift-surfaces used by operational, experimental, and historical aircraft. In the image/animation above, the profiles are sorted by a close approximation of aerofoil thickness and overlayed. The aerofoil data varies a lot in that there is little to no consistency in scale, proximity of leading-edge datapoints to the origin, rotation and so on, so each dataset had to be transformed. All operations, plotting and animations were made using MATLAB r2018a.
There are a few weird shapes in there: rotor blade ends, cowls, primitive Wright Brother aerofoil shapes; but predominantly there is an obvious popularity in the conventional symmetric tear-drop shape.
The above image is an attempt at generating the 'average' aerofoil profile: created by recreating a matrix of airfoil y values interpolated to to a more uniform x domain of 64 points, and then calculating the average y-coordinate for each point along the horizontal axis. This seems to be a clunky method as the resulting outline isn't exactly very smooth, and results seem prone to some variance when increasing or decreasing the number of points we condense or extrapolate each set of coordinates to - however, 64 is within a stable range. Surprisingly the lower edge of the average aerofoil is very flat - this is probably influenced by aerofoils with a more significant camber.
This is a v/V flowfield plot of the (slightly smoothened) 'average' aerofoil, simulated in freestream airflow at Mach 0.5 between angles of attack: -5 to +20 degrees, with iso Cp lines and streamlines overlayed.
Spoiler: Meh, it's ok.
So, lesson, I guess: if you're designing an aerofoil, even though a lot of the process in industry is reliant on legacy: don't just do the average based on what everyone else is doing - design specifically for your requirements.