SIGNAL AND NOISE

Let's say you're tuning into a radio station.  If the station is
local, you can hear the music fine.  But if the station is far
away, there may be some static.  If there's lots of static, it
drowns out the music.

Now let's say you're trying to measure the brightness of a planet.
If you're on a dark mountaintop in Arizona, you get very little
stray light; but if you're doing it in Times Square, all kinds of
non-planetary light bombard you from all directions.  Exactly the
same situation as the distant radio station.

The light you measure from the planet is your SIGNAL.  The stray 
light detracts from your ability to measure the planet's brightness
accurately, and contributes to the statistical NOISE.  

Once again, if you're counting, say photons of light in this case,
you can use Poisson statistics.  The signal is the number of counts
coming from the object you want to measure.  The noise is the square
root of the total number of counts you collect, from both the object
and stray sources.  

For example, on the mountaintop you may measure 100 photons from
the planet in one second, and zero photons of stray light.  Then
the signal is 100, and the noise is sqrt(100).  Then the margin of
error of this measurement is 

   sqrt(100)
   --------- = 10%
      100

The quality of a measurement can be expressed by the ratio we call
SIGNAL-TO-NOISE, which is exactly what it sounds like:

   S     100                                                    N
  --- = -------- = 10  [ note that the margin of error is just --- ]
   N    sqrt(100)                                               S

Now, say you make the same measurement in Times Square.  Now you get
100 photons from the planet, plus 500 photons from the neon lights.
In this case, the signal is still 100, but the noise is sqrt(100+500),
or sqrt(600).  Now the margin of error is

   sqrt(600)                                        100
   --------- = 24.5%  and the signal-to-noise is  --------- = 4.1
      100                                         sqrt(600)

These numbers make common sense; you'd expect a measurement made in
Times Square to be less accurate than one on a dark mountaintop, right?