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Here's how to figure it.
The diameter consists of two sidewalls plus the wheel diameter, or
D = 2S + W, where
S= Sidewall Height = Cross-sectional width x aspect ratio
W= Wheel diameter
Since tire size is a mix of inches (wheel diameter) and millimeters (tire section width), you have to convert to a single system of units. I suggest mm.
For the 225/55-16 tire:
Section Height = 225mm x 0.55 = 123.75mm
Wheel diameter = 16in x 25.4mm/in = 406.4mm
Total diameter = (123.75 x 2) + 406.4 = 653.9mm
For the 195/65-15 tires,
(195 x 0.65 x 2) + (15 x 25.4) = 253.5mm + 381mm = 634.5mm
The new tires (16in) are 19.4 mm larger in diameter, an error of 3%, causing the speedometer to read 3% slower (e.g., 1.5 mph less at 50 mph) and your odometer to read 3% fewer miles (e.g, 3 miles less for every 100 miles driven, or 30 miles for every 1,000 miles driven).
Whether this is significant for you is your personal decision.
As a rule of thumb for a plus one conversion (going up one inch in wheel diameter) without a change in diameter, you drop "10" in aspect ratio number and go up ten in section width. So from a 195/65-15 tire, you would go to a 205/55-16 tire with the approximately same overall diameter:
e.g.: (205 x 0.55 x 2) + (16 x 25.4) = 225.5mm + 406.4mm = 631.9mm
which is almost identical to the original tire, only 2.6mm different (less than tread thickness variations) or 0.4%.
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