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Physics 114: Lecture 8
Measuring Noise in Real Data
Dale E. Gary
NJIT Physics Department
Mean and Standard Deviation
1
N
x

Sample Mean x 

1
Parent population mean   lim 
N  N


Standard Deviation from sample mean

2
1
Standard Deviation from parent population mean   lim    xi     .
N  N


i

x
 i 
2
 1
s 
 xi  x   .

 N 1

February 12, 2010
Homework 1 Data
Note, “data” is plural
Hat-P-6 b Transit
10.44
10.45
10.46
V Magnitude
10.47
10.48
10.49
10.5
10.51
10.52
10.53
3
3.5
4
4.5
Time (UT hours)
20
Number of points
The HAT-P-6 b transit data are
shown at the right.
 If in MatLAB you type
mean(a(:,6)) and std(a(:,6)), you
will find that the data have a
mean of 10.50, and standard
deviation of 0.015.
 The plot at lower right shows the
histogram of the measurements
with an overlay of a Gaussian
(normal distribution) bell curve
using the parameters above.

15
10
5
0
10.44
10.45
10.46
10.47
10.48
10.49
V Magnitude
10.5
10.51
10.52
February 12, 2010
10.53
Homework 1 Data
As an example of evaluating data in a real application, consider the HAT-P-6
data from homework 1.
 This is data taken during an eclipse of a star by a planet (that is, the planet is
crossing in front of the star, causing a very small decrease in light level).
Unfortunately, I could not get everything set up in time, and I only got the
time at the end of the eclipse (egress).
 The data came from images of the star field, and there are several steps to
obtaining the light curve.

Two examples of
eclipses by others,
with more complete
lightcurves.
February 12, 2010
Homework 1 Data
Here is a fit to the
measurements that you read
in. The curve is the
expected eclipse lightcurve
obtained from “forward
fitting” using a model for the
eclipse.
 Note the “trend removed”
curve, which is an example
of a systematic error.

February 12, 2010
Homework 1 Data
The magnitude measurements are themselves made with images from a CCD
camera, which have their own systematic and random errors.
 The systematic errors can be removed through calibration, and as mentioned
before, they include both additive
and multiplicative errors.
 To remove such systematic errors,
we want to make the random errors
in the calibration data as small as
possible.
 Let’s go through the process and
introduce CCD cameras.

February 12, 2010
How CCDs Work
• Photons to Analog/Digital Units (Counts)
m
m
One photon has 73%
chance to cause release
of an electron (e-). It takes
1.6 e- to give 1 count. So
100 photons will result in
100*0.73/1.6 = 45 counts.
Each well can hold
120,000 e- = 55000 counts
These 2 parameters
give conversion of
photons to counts
2010 Feb 13
How CCDs Work
• Bias (additive)
m
m
Even with 0 s exposure,
just reading out the image
gives (on average) 17 e-,
or about 10 counts. This is
called bias, and is neither
temperature nor time dep.
These 2 parameters
give noise output
2010 Feb 13
How CCDs Work
• Dark Current (additive)
m
m
With a time exposure,
say a 1 min exposure at
-30 C, will have 19 more
counts. This is BOTH
temperature and time dep.
These 2 parameters
give noise output
2010 Feb 13
Imaging First Principles
•
•
•
•
•
The last step is to take calibration frames: Bias, Dark, and Flat frames.
I take 20 Bias and 20 Dark (set camera cooler to temperature first, and take
dark frames for same duration as imaging frames). I take 10-20 flat frames
(need even illumination—set duration for mid-range exposure).
Bias frames are instantaneous, for subtraction of read noise.
Dark frames are same duration as imaging frames, for subtraction of dark
current and correction of hot pixels.
Flat frames are for removal of non-uniform illumination (vignetting and dust).
Images are divided by flat frames.
2010 Feb 13
Imaging First Principles
•
Noise is the enemy, so average calibration frames.
2010 Feb 13
Imaging First Principles
•
Flat field light box
Image
calibration
Image without
with Calibration
2010 Feb 13