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Noise is fairly evenly distributed but it will be more significant relative to the small amount of data in the dark areas of a scene, whether they be dark subjects, brighter subjects in shadow, or just underexposed more than necessary. To counter the effect of such noise you can apply noise reduction in post processing but to do the best job you should increase the exposure as much as you can without losing required highlight details due to overexposure, and then to restore balance in post processing you would reduce exposure by an appropriate amount. This process is called ETTR (expose to the right) but the process to the left part is usually not mentioned.
ETTR has some inherent difficulties. In no particular order they include...
1. How do you know how far to the right to go ? Without experience you rely on a histogram that will easily lead you astray.
2. The histogram is based on a converted jpg version of the raw data.
3. That converted jpg uses all of the camera settings such as sharpening, contrast, picture style, saturation, noise reduction and anything else that affects the pixels after they have been captured.
4. The histogram is highly compressed at the dark and light ends of the scale and you'll be lucky to see 5 stops of useful data from a raw file that might contain 12 or 14 stops of dynamic range.
5. The part of the histogram that most affects whether or not you can apply more exposure may only be 1 pixel wide and has a poor sense of scale for indicating how much of the image is affected
6. If there is a big gap in brightness between the highlights that you want to keep and the rest of the scene then the histogram is borderline useless.
7. You can affect the histogram and the preview image by changing the picture style to neutral, the contrast to low, the sharpening to minimal, etc. The preview will look crappy but you'll get a better idea of how much of the scene is really close to overexposure because a lot of what was falling off the edges of the histogram will be brought towards the centre of the histogram.
8. You need to pay attention to the colour histograms as well as the luminance histogram because it is very easy to blow a lot of detail in say red or blue channels without having any pixels turn white, so the luminance histogram will not indicate overexposure.
9. ETTR might even become ETTL in some cases where a reduction in exposure is needed to preserve some bright data.
10. Sometimes, perhaps often, you just can't capture all of the DR that you want. Either decide what to lose or bracket exposure over several shots so that you can sort it all out later on in processing.
A long time ago (with which camera I cannot recall) I did some exposure tests and checked the histograms. I found that with low contrast settings I could get an extra 1/3 of stop of highlight detail out of a raw file compared to what the histogram showed, but with high contrast settings in the camera I could recover 4/3 stops of highlight detail. So then I realised that I had four basic options regarding the histogram and the preview image:
1. use camera settings that best showed a nice looking preview and try to deal with the exposure limits later on.
2. use a high contrast setting to force bright areas into overexposure on the histogram so that I got lots of warning that real loss of highlights was imminent. I still had 4/3 stop of recovery but that may or may not be enough.
3. Use a low contrast setting and treat the overexposure warnings differently - as an indicator that overexposure was almost certain because it was almost too late to recover it. The good new was that more of the real data was in the more useful part of the histogram.
4. Largely ignore the histogram and bracket exposures for everything that was important enough to photograph.
These days I use options 2, 3 and 4 depending on the subject and how easily I can retake a poorly exposed photo.
If your 5D is consistently overexposing then you need to determine whether that is based on pixel values or simply what you see on the screen. You also need to pay attention to those camera settings that affect the histogram, and if necessary choose something different. I don't think the camera meter pays much attention to those camera settings because it sees the scene directly before the sensor does, so there may be some discrepancy between the metered exposure and the histogram that is based on processed sensor data.