Most cases? I would say in the case of the same brand/model line.
Dollar for dollar? Depends.
Current situation, or in this case:
Competing brands at similar price-range(10"vs12")? Depends. If the 10" is the same price as the 12", the 10" may be of better quality, and have much more power.
Competing brands where the 10" is in a higher class/price-range. Probably not.
I know what you mean. And, you know what I mean. But, for many, they will see this and assume that in "most cases" the 12" will outperform the 10". That is just not fact.
Within the same product line it is almost always accurate. From brand to brand I would say it is accurate about 60%-70% of the time perhaps more. When you are talking a $1,000 10" vs. a $250 12" that is not a common sense comparison and will result in highly skewed results.
The other thing that comes into play is the enclosure design. One must compare sealed to sealed and ported to ported.
More power does not mean lower frequency response. Either a 10" has sunstantially more excursion capability to allow it to play deeper than a 12" in a sealed application or it is tuned lower in a ported application (and could also have higher excursion capability). Otherwise the 12"
should trump the 10" in low end response nearly every time. It is simply more efficient at playing a lower frequency.
So realistically, yes, it is a fact
if the drivers themselves are compared with other variables being equal. Note the Fs on most subs especially those from the same manufacturer in the same series, the larger the woofer the lower the Fs.
Here is an example. Let's say two woofers, a 12" and a 10", from the same manufacturer and same series are playing a 30hz note at 100db. The 10" woofer is playing at 95% of it's excursion capability while the 12" is at merely 75%. If we lower the frequency to 20hz the 12" will become louder as the 10" will no longer be able to increase in volume as its excursion threshold is reached. The 12" may now be at 100% of its linear capability (at 20hz) while the 10" would need to be at 125% of its linear capability to keep up resulting in much distortion. The power output would need to be backed off to keep the 10" at its linear threshold resulting in lower output. It is not saying the 10" cannot play 20hz, it is just that the output, if kept linear, is not nearly as useable as the 12". Therefore the 12" "plays deeper".
What I am saying is not that the 10" driver cannot produce the low frequencies, it is just you get more useable low frequency response with a larger driver due to its increased efficiency at all frequencies.
I hope this makes some sense and I did not create more confusion. Sometimes I have a tendency to do that.