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There seem to be divergent opinions on the correct diameter of 16 gauge.

Boxall and Edmiston feels that it is .670"
http://www.boxallandedmiston.co.uk/shotg...ghtly-different

Wikipedia has it as .663"
https://en.wikipedia.org/wiki/Gauge_(bore_diameter)

Hallowell at .662"
http://www.hallowellco.com/bore_size_chart.htm

World Almanac and Book of Facts at .660"
https://www.unc.edu/~rowlett/units/scales/shotguns.html

International Hunter Education Association at .670"

forensicmed.co.uk at .665"

And so on.

If you look at the British proof house charts they list 16 gas as .662". "IF" you look up the weight of lead & calculate the diameter you will find that every gauge in the British system is "Just a Wee Bit Off". Calculations based on the weight of lead gives a very slightly larger diam than the charts. My "Assumption" is they took some given weight of pure lead, quite possibly a pound, formed it into a sphere & measured the diameter & calculated the rest. Perhaps in forming the sphere they compressed the lead just a bit & then used this size as the basis for the rest. By mathematical calculations .662" for the 16ga "Fits" in with all other gauge sizes from 1 to 172.28 in both English & German charts. Call it what you will, but if .662" is incorrect then ALL other charted gauge diameters are also incorrect.
One thing is for certain, Gunmakers have not always adhered to the standard. As noted for many years L C Smith bored 16 gauge barrels to "Their" standard of .650", others have also varied. None of this affects the true standard based on that 16 pure lead balls with a diameter which just fits the bore weigh 1 pound. Off size balls do not fit that standard.

Jack O'Connor's "Shotgun Book" (1965) gives 16ga bore diameters from the following manufacturers:
Winchester: .664
Browning: .665
Remington: .673.

I'd note that the same book shows a difference of .008 between standard 20ga bore diameters, and .005 between standard 12ga BD's.

Brophy's L.C. Smith book (with factory drawings) shows both Miller's .650 figure for the 16 as well as .652. Either case, a long ways from Remington's .673!.

Us guns are usually built to SAAMI specifications, and current SAAMI specs call for a bore diameter of 0.665" with a tolerance of 0.020", so a modern gun could have a bore of .665" to o.685" and still be within industry specs. Having said that, I have always heard that the 16 ga bore diameter was 0.662".

Once the diameter of a 1 pound lead ball (1 gauge) is known it is a simple matter to find the diameter of any gauge. You simply cube the 1 gauge size, divide by the gauge desired & then take the cube root of the result. One day some years back at work I was running a CNC milling machine on some airfoils which had cuts running for several hours each. I had to stay with the machine but had virtually nothing to to unless some unexpected event were to occur. I had copied off the chart of British proof gauge sizes. I had a programmable Casio Calculator. After putting in the program using a given size for 1 gauge I only had to enter a different gauge & it would give me the answer.
I very quickly discovered that using 1.669" for the base that all sizes did not fit, some came out too small. Stepping up to 1.6695 some sizes cam out too large. It proved out that to get a perfect match for the charts the 1 gauge ball had to be carried out to 6 decimal places. There was a very narrow range which would fit by rounding the end result to 3 decimal places. The figure I settled on was 1.6699285" for the 1 gauge ball.
As stated using this diameter fits all gauges listed in British, German & I believe Belgian charts from the proof houses from 1 gauge down to 172.28 gauge (.300 caliber).
These sizes have been accepted as standard since well back into the 1800's at least. Using .665" for the 16 would give .732" for the 12. It has always been listed as .729". Sizes for the common gauges are;
10=.775"
12=.729"
16=.662"
20=.615"
28=.550"
These sizes all have a perfect fit to "The Formula".
I also discovered that to obtain the "Tween" gauges as in 13/1 you average 12 & 13 then drop to three decimal places, the tween sizes are not rounded as are the primary gauges. On the larger gauges with two tween sizes as 8/1 & 8/2 you use third & two thirds & drop beyond three decimal places.
I have absolutely No Idea as to why SAAMI opted to change the size of the 16 but the .662" size has been nominal for "Well Over" a century.
As has been noted though individual makers have always bored guns to whatever they desired. The proof houses used the actual bore size to determine the proof charge, not the chambering. Thus if you had a gun chambered 12ga, but bored for brass shells with an enlarged bore which came up to 10 gauge diameter the gun would be proofed as a 10, not a 12.

Miller: the current SAAMI specs vary from the traditional sizes for the 12, 16, and 28 gauges, while the 10, 20, and .410 remain unchanged. The current sizes are 0.725" for the 12, 0.665" for the 16, and 0.545" for the 28, so two are smaller than the traditional sizes, and one is larger. I did not find an explanation for the variations.

If that tolerance works in both directions, then the bore diameter could be anywhere from .645-.685.

Tom;
Quite obviously SAAMI has not followed the traditional formula for gauges. Sort of makes the word Gauge of no effect as it would seem to be based on Nothing. To truly be a Gauge strongly implies a standard.

Larry: The tolerance is a one sided tolerance. There is a 0.020" on the plus side for all bore sizes, but no minus tolerance.

According to my (Rather Old) Machinery's Handbook they give the weight of lead @ 707.7 lbs per cubic foot. Using this figure a one lb ball would have a diameter of 1.670701 to six decimal places. This would make;
12 ga = .730"
16 ga = .663"
20 ga = .615"
28 ga = .550"
Thus two sizes show .001" larger than the charts, two remain unchanged. Some other gauges could grow by from .001"-.002" while many would stay the same, none would "Shrink" as the base for calculating (1 gauge) is larger by 1.416".
This is probably more than anyone wanted to know, but this is the Nitty Gritty in a Nutshell on gauge sizes.