Because CCDs and CMOS imaging chips are virtually linear devices within their useful range, and because the analog-to-digital converters that digitize their outputs are also linear, the signal-to-noise ratio is low for darker regions of the tone curve (the left side of the histogram). By exposing as far to the right as we can without clipping any highlights, we can achieve the higher signal-to-noise ratio possible in the final image. That means we will have the least possible amount of noise, visible artifacts, color errors, and banding.
Here are some details:
On a pixel-for-pixel basis, the sensor technology being equal, the signal to noise ratio (SNR) is proportional to the square root of the photosite area. This means the SNR is proportional to the pixel pitch.
It’s not really fair to compare noise in cameras with wildly varying resolution. Fortunately, we don’t have to. Consider two cameras using full frame sensors with the same technology, one 40 megapixels, and the other 10 megapixels. The sensor with more pixels will have half the SNR of the sensor with fewer, if the measurements are performed under the same conditions. If we res down the 40 megapixel to 10 megapixels, to a first approximation it will have the same SNR as the 10 megapixel camera. So, technology and output resolution held constant, the SNR of a camera is proportional to the linear dimensions (length, width, or diagonal – your choice, if the aspect ratio is the same) of the sensor.
So, if a full frame camera, measured under a standard set of conditions with its output file res’ed to a certain resolution has an SNR of x, an APS-C camera will have an SNR of 0.7x. A micro four-thirds camera will have an SNR of half of x. A Leica D-Lux 4 will have an SNR of one quarter x. An iPhone will have an SNR of one-eighth of x. Going the other way, a medium format camera will have an SNR of somewhat less than 2x, and a 4×5 scanning back will have an SNR of a little less than 4x.
Can we quantify the SNR effects of ETTR? Indeed we can. Let’s take an image that’s exposed perfectly to the right. No clipping or blown highlights, but information in the very top histogram bucket. Let’s pick a pixel group in that image, and measure its SNR. Let’s say it measures y. If we underexpose one stop from the perfect ETTR image, that pixel group will have an SNR of 0.7y. If we’re two stops under, the SNR is half of y. Four stops under, and it’s an quarter.
So, from a noise point of view, you can turn your full frame SLR into a micro four-thirds camera by underexposing by two stops, into a point-and-shoot by underexposing four stops, or into an iPhone by underexposing by six stops. Admittedly, I’m painting with a broad brush. The sensor technology in the iPhone is probably different from that in a D4 by more than just geometry. Still, it’s a useful way of looking at ETTR.