Does my (too low) tire pressure, affect the speedometer of my car? Does my (too low) tire pressure, affect the speedometer of my car?
How big is this effect?
 A: No. Automobile tires to not expand radially to any great extent - the steel belts will keep that from happening. So, the tire radius still determines how far the car travels per rotation. Now, if your tires are slipping on the road, or are slipping with respect to the rims, than yes you have speedometer problems, but you have lots of other problems as well...
A: Yes.  With low air pressure, the axle is closer to the road:  the radius of rotation is reduced.   Some cars have systems that warn you when your tire pressure is low.  The way they work is by measuring the rotation rate of the tires.  If the rotation rate exceeds a certain limit, the dashboard light glows.   With under-pressured tires, your speedometer will display a value greater than your actual speed.
If your tire is under-inflated to the extent that the axles have sunk say, one inch or so closer to the ground, the error would be roughly 5% or so.  (Depends on the size of the tires).
A: The mechanism of a speedometer determines the travel speed of the vehicle indirectly by measuring the rotation speed of the transmission output or wheels. One turn of the wheel causes the vehicle to cover a distance equal to the circumference of the wheel, so there is a direct relation between the wheel's angular speed and the vehicle's linear speed; however, since a smaller wheel has to turn more times than a larger wheel to cover the same distance, the size of the wheel effects the accuracy of the speedometer.
Let's assume that at optimal pressure your speedometer is exactly accurate. The wheels have a radius $r$ and a circumference $2\pi r$. That circumference is what the speedometer is "expecting" when it displays your current speed. Now we remove enough air that the effective wheel radius (the distance from the axle to the ground) is reduced by ten percent; the new radius is $0.9r$ and the new circumference is $1.8\pi r$, also a ten percent reduction since circumference is linearly related to radius.
A vehicle traveling on the filled tire could cover a certain distance $x$ in 100 turns of the wheel. However, the underpressurized tire is 9/10ths the size of the filled one, and thus has to turn 10/9ths as many times to cover the same distance. So if the vehicle is traveling at 50km/h on the low tires, the speedometer will report slightly over 55km/h.
In summary, the speedometer reports higher-than-actual speeds for low (or smaller) tires, and lower-than-actual speeds for larger tires, and the proportion of difference is the reciprocal of the difference in tire size from "normal."
A: Looking at tire specifications from tirerack.com for tire size 185/65/15 .
this tire has a diameter of 24.4 inch but rotates only 852 revolutions per mile.
The circumference calculated should rotate 833.39 rev per mile. Suggesting air pressure and tire squish affects distance per revolutuon? 
While tirerack has about 20 tires from various manufactures in this size 185/65/15
99% are 24.4 inch diameter one is 24.5 inch
while most specify rotations at 852 rev per mile three are 846,848,865.
I purchased pirelli 185/65/15 tires my speedometer error is twice as much as it was with japanese goodyear integrity tires while both tires spec at 852 rev per mile. While the pirelli's sidewalls are balloned out the goodyears sidewalls were not balloned out but were straight verticle. The pirelli have a soft cushioned ride compaired to the straight sidewalled goodyears. The axle distance to road (tire radius) is much less with the pirelli's. I'm attributing this different radius to the increased speedometer error. My car is prius genII.
