I am designing the instrument. I am exploring the number of samples
required to reduce the effect of chance on the measurement result
(in
respect of the sampling issue) to an acceptable figure.

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The first thing to do is calibrate the instrument against a standard
noise source. Immediately, the uncertainty in the standard is
transferred to the instrument - plus some more uncertainty due to the
manner in which the standard and instrument are associated.

Does the instrument read in watts, decibels, or what?

The second thing to do is to be verbally and numerically more precise
about "to reduce the effect of chance on the measurement result to an
acceptable figure."

At the outset you should define the acceptable figure. What effects?
In what units is the acceptable figure?

It is then not a difficult matter to decide the number of
measurements, by taking samples, to give a predetermined level of
confidence in the average or mean. But I have the feeling you are
over-flogging the issue. You don't really have a problem.
----
Reg.