
The same excerpt of a 10 minutes flat, recorded by the complete equipment
(telescope + Lhires III). The pixels are saturated to about 30% (16 Bit,
max. ADU = 2 ^ 16 = 65536). There exists a clear gradient . The hot pixels
are visible, of course, they also contribute their increased rate of darkcurrent.
The normal "noise" is about 150 ADU (excluding hot pixels). The S / N
in the flat can be roughly estimated at 20,000 ADU to 20000/150 = 133.
Subtracting from the flat the dark, is in the resulting masterflat even
the 10 ADC noise of the dark introduced.

Midas 001> STATIS/IMAG bias001R.fit
frame: bias001R.fit (data = UI2, format = FITS), complete area of frame
minimum, maximum: 0.000000e+00 1.830000e+02 at pixel (1,1),(185,81)
mean, standard_deviation: 8.912154e+01 8.233002e+00
3rd + 4th moment: 5.60833 51.7889
total intensity: 8.68133e+06
median, 1. mode, mode: 8.914917e+01 3.588235e01 9.006470e+01
total no. of bins, binsize: 256 7.176471e01
# of pixels used = 97410 from 1,1 to 382,255 (in pixels)
Midas 002> STATIS/IMAG dark011R.fit
frame: dark011R.fit (data = UI2, format = FITS), complete area of frame
minimum, maximum: 0.000000e+00 3.908300e+04 at pixel (286,17),(325,184)
mean, standard_deviation: 1.357749e+02 3.368284e+02
3rd + 4th moment: 54.0146 4278.68
total intensity: 1.32258e+07
median, 1. mode, mode: 8.455499e+01 7.663333e+01 7.663333e+01
total no. of bins, binsize: 256 1.532667e+02
# of pixels used = 97410 from 1,1 to 382,255 (in pixels)
Midas 003> STATIS/IMAG flat001R.fit
frame: flat001R.fit (data = UI2, format = FITS)complete area of frame
minimum, maximum: 5.190000e+02 5.573700e+04 at pixel (369,1),(74,238)
mean, standard_deviation: 1.160649e+04 1.106488e+04
3rd + 4th moment: 0.432029 1.74744
total intensity: 1.13059e+09
median, 1. mode, mode: 1.236926e+04 8.438118e+02 8.438118e+02
total no. of bins, binsize: 256 2.165412e+02
# of pixels used = 97410 from 1,1 to 382,255 (in pixels) 
The text on the left shows the printout of a command in MIDAS. There
are calculated the image statistics for the images analyzed above. Important
are the average (mean) and the standard deviation of pixel intensities.
The mean (S) of the bias is 89.1 ADU with a standard deviation
of 8.23 (N). The median S / N in the bias is therefore 89.2 / 8.23 = 10.8.
That corresponds fairly closely to the expectation for a random distribution.
In such a case, the standard deviation (= noise) ~ squareroot (signal).
In the dark, of course, is the mean value increased on 135.8 caused by
the warm pixels and the dark current, compared with the above <1s exposed
bias. The standard deviation is much higher (336.8). However, during the
actual data reduction, the effects of dark current and hot pixels is eliminated
by the dark subtraction from the object images.
It is important to look at the flats. The exposure time is set such that
the pixels are saturated to 30 to 50%. In the flat but in general, structures
are found: the shadow of dust grains on the glass front of the CCD chip,
horizontal dark streaks caused by dust in the slit, vignettings... Therefore,
the standard deviation over all pixels of the flat is very high. This
standard is not synonymous with noise but is the result of the depicted
structures. The noise has only a small proportion in the standard deviation,
roughly in the magnitude of the darks.
As shown above, is the noise in the investigated section of the flats
pixel rows around 150. Theoretically, expected to be root (20000) = 141,
ie practically the same value.
The Midas command 'statistics / ima' yields for the flat: mean = 11,606,
standard deviation 11065 ADU.

IMPORTANT: By any correction using biases or darks or flats additional
statistical errors are introduced into the resulting spectrum. If Ni is
the noise in image i, and N is the noise in the sum spectrum, the mean
square of all is (in the sum of n involved single shots + darks + flats):
N = squareroot ( N1 ^2 + N2 ^2 +...Nn ^2)
A sum spectrum, which is composed of one or a few long exposed pictures,
provided a better (higher) S / N as a sum spectrum, which is integrated
from many short exposed images (same total exposure time). Each recording
smuggles its own noise in the sum spectrum. This noise in each individual
recording is also genereated by the electronic noise (readout noise, etc.),
which is the photon noise superimposed. Scientifically evaluable are spectra
whose S / N is over 100. For analysis of time series even S / N up to
1000 is desirable. I target, whenever possible  in general, an S / N
of > 200.
