The only advantage of synthetic oils is their ability to withstand greater heat, and breakdown slower than dino oils. Also, additive packages may be a little better in most synthetics sinve they offer longer change intervals. Any MPG improvements would be very small and hard to prove it was the oil or some other insignificant factor.
If you change your oil religiously every 3000-5000 miles, you will probably never see any difference. Even though I would normally change conventional oil every 3000 miles, and with Mobil-1 I go for 5000 miles, and if that slides to 6000, or even 7000 miles, I don't have to worry that something will fail.
As for Royal Purple, Amsoil, and many other's, they are also synthetics. I do know that Royal Purple ran a test by changing all the fluids to Royal Purple (Engine oil, Transmission oil, and gear oil) products and increased mileage by 2-3 MPG and picked up and additional 8 HP. All were due to friction and heat reduction using their lubricants.
...Rich