The error and the standard deviation are sometimes confused with each other. Therefore, before continuing with the various measurement limits below, let’s review these two fundamental concepts. Instead of going into the theoretical definitions of the two (which you can see in their respective Wikipedia pages), we’ll discuss the concepts in a hands-on and practical way here.
Let’s simulate an observation of the sky, but without any astronomical sources! In other words, where we only a background flux level (from the sky emission). With the first command below, let’s make an image called 1.fits that contains \(200\times200\) pixels that are filled with random noise from a Poisson distribution with a mean of 100 counts (the flux from the background sky). Recall that the Poisson distribution is equal to a normal distribution for larger mean values (as in this case).
The standard deviation (\(\sigma\)) of the Poisson distribution is the square root of the mean, see Photon counting noise. With the second command, we’ll have a look at the image. Note that due to the random nature of the noise, the values reported in the next steps on your computer will be very slightly different. To reproducible exactly the same values in different runs, see Generating random numbers, and for more on the first command, see Arithmetic.
$ astarithmetic 200 200 2 makenew 100 mknoise-poisson \
--output=1.fits
$ astscript-fits-view 1.fits
Each pixel shows the result of one sampling from the Poisson distribution. In other words, assuming the sky emission in our simulation is constant over our field of view, each pixel’s value shows one measurement of the sky emission. Statistically speaking, a “measurement” is a sampling from an underlying distribution of values. Through our measurements, we aim to identify that underlying distribution (the “truth”)! With the command below, let’s look at the pixel statistics of 1.fits (output is shown immediately under it).
$ aststatistics 1.fits Statistics (GNU Astronomy Utilities) 0.22 ------- Input: 1.fits (hdu: 1) ------- Number of elements: 40000 Minimum: 61 Maximum: 155 Median: 100 Mean: 100.044925 Standard deviation: 10.00066032 ------- Histogram: | * * | * * * | * * * * | * * * * * | * * * * * | * * ******** * * | * ************* * | * ****************** * | ************************ * | ********************************* |* ********************************************************** ** * |----------------------------------------------------------------------
As expected, you see that the ASCII histogram nicely resembles a normal distribution. The measured mean and standard deviation (\(\sigma_x\)) are also very similar to the input (mean of 100, standard deviation of \(\sigma=10\)). But the measured mean (and standard deviation) aren’t exactly equal to the input!
Every time we make a different simulated image from the same distribution, the measured mean and standard deviation will slightly differ. With the second command below, let’s build 500 images like above and measure their mean and standard deviation. The outputs will be written into a file (mean-stds.txt; in the first command we are deleting it to make sure we write into an empty file within the loop). With the third command, let’s view the top 10 rows:
$ rm -f mean-stds.txt
$ for i in $(seq 500); do \
astarithmetic 200 200 2 makenew 100 mknoise-poisson \
--output=$i.fits --quiet; \
aststatistics $i.fits --mean --std >> mean-stds.txt; \
echo "$i: complete"; \
done
$ asttable mean-stds.txt -Y --head=10
99.989381 9.936407
100.036622 10.059997
100.006054 9.985470
99.944535 9.960069
100.050318 9.970116
100.002718 9.905395
100.067555 9.964038
100.027167 10.018562
100.051951 9.995859
100.000212 9.970293
From this table, you see that each simulation has produced a slightly different measured mean and measured standard deviation (\(\sigma_x\)) that are just fluctuating around the input mean (which was 100) and input standard deviation (\(\sigma=10\)). Let’s have a look at the distribution of mean measurements:
$ aststatistics mean-stds.txt -c1 Statistics (GNU Astronomy Utilities) 0.22 ------- Input: mean-stds.txt Column: 1 ------- Number of elements: 500 Minimum: 9.98183528700191e+01 Maximum: 1.00146490891332e+02 Mode: 99.99709739 Mode quantile: 0.49498998 Median: 9.99977393190436e+01 Mean: 99.99891826 Standard deviation: 0.04901635275 ------- Histogram: | * | * ** | ****** **** * * | ****** **** * * * | * * ************* * * | * ****************** ** | * ********************* *** * | * ***************************** *** | *** ********************************** * | *** ******************************************* ** | * ************************************************* ** * |----------------------------------------------------------------------
The standard deviation of the various mean measurements above shows the scatter in measuring the mean with an image of this size from this underlying distribution. This is therefore defined as the standard error of the mean, or “error” for short (since most measurements are actually the mean of a population) and shown with \(\widehat\sigma_{\bar{x}}\).
From the example above, you see that the error is smaller than the standard deviation (smaller when you have a larger sample).
In fact, it can be shown that this “error of the mean” (\(\sigma_{\bar{x}}\)) is related to the distribution standard deviation (\(\sigma\)) through the following equation.
Where \(N\) is the number of points used to measure the mean in one sample (\(200\times200=40000\) in this case).
Note that the \(10.0\) below was reported as “standard deviation” in the first run of aststatistics on 1.fits above):
$$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{N}} \quad\quad {\rm or} \quad\quad \widehat\sigma_{\bar{x}}\approx\frac{\sigma_x}{\sqrt{N}} = \frac{10.0}{200} = 0.05$$
Taking the considerations above into account, we should clearly distinguish the following concepts when talking about the standard deviation or error:
This is the standard deviation of the underlying distribution (10 in the example above), and shown by \(\sigma\). This is something you can never measure, and is just the ideal value.
Ideal error of measuring the mean (assuming we know \(\sigma\)).
Measured Standard deviation from a sampling of the ideal distribution. This is the second column of mean-stds.txt above and is shown with \(\sigma_x\) above. In astronomical literature, this is simply referred to as the “standard deviation”.
In other words, the standard deviation is computed on the input itself and MakeCatalog just needs a “values” file. For example, when measuring the standard deviation of an astronomical object using MakeCatalog it is computed directly from the input values.
Measurable scatter of measuring the mean (\(\widehat\sigma_{\bar{x}}\)) that can be estimated from the size of the sample and the measured standard deviation (\(\sigma_x\)). In astronomical literature, this is simply referred to as the “error”.
In other words, when asking for an “error” measurement with MakeCatalog, a separate standard deviation dataset should be always provided.
This dataset should take into account all sources of scatter.
For example, during the reduction of an image, the standard deviation dataset should take into account the dispersion of each pixel that comes from the bias, dark, flat fielding, etc.
If this image is not available, it is possible to use the SKY_STD extension from NoiseChisel as an estimation.
For more see NoiseChisel output.
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GNU Astronomy Utilities 0.22 manual, February 2024.