In astronomy almost every quantity we observe has an associated error often
expressed as a $ \pm $ quantity after the value, ie:

$ M = 4.1 \pm 1.0 M_\odot $

suggests that our best estimate of the mass $ M $ is 4.1 M$ _\odot $ but that
it could easily be 1.0 M$ _\odot $ more or less than that.

Sometimes astronomers choose to drop the $ \pm $ and just put the error in the last digit in parentheses (brackets). So in the example above this would be somewhat confusingly written as:

$ M = 4.1(10) M_\odot $

Novices sometimes think this means $ 4.1\pm10.0 $ – it does not!

More awkwardly, if the best estimate was $ M=4.0 M_\odot $ then we could have written:

$ M = 4(1) M_\odot $

which is equivalent to

$ M = 4.0(10) M_\odot $!

Normal or Gaussian Distributions

In nature when we make a series of measurements they
often follow a Gaussian or Normal distribution like that shown above.

What does this really mean? Well, astronomers usually set the error to mean there is a 67% chance that the true value falls within one listed error of the value. If the measurements are distributed “normally”, ie in a Gaussian fashion, then there is about a 96% chance the value is within twice the listed error and a >99% chance it is within three times the error. The error is often referred to as “sigma”. So a 1-sigma result is quite poor, whereas a 3-sigma result reasonably secure. The sigma comes from the standard deviation of a Gaussian distribution.

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