A/D conversion improvement through Arduino
As Tom Carpenter has already pointed out, you must divide by 2^N = 1024.
You are correct with your second assumption. You always want to fit the ADC full scale voltage (the maximum voltage it can measure) with your maximum input amplitude. Typically there are two ways to do that:  


You can use an input amplifier to amplify/attenuate your signal to exactly match the FS voltage.  
You can set the reference voltage of the ADC to the desired value.



While the controller used on the Arduino Uno (Atmega) offers in general the option to connect an analog reference voltage differing from general Vcc, this does not work for you, because the analog reference input is hardwired on the Arduino to 5V. So you are stuck with a reference voltage of 5V, which means your FS (full scale voltage) will always be 5V on the Arduino.
What you CAN do, is to use an operational amplifier to increase your signal's amplitude by factor of 2.5 from 2V to 5V. That way, you get the full resolution of the 10 bit ADC.