9.5. Measurement Uncertainty 549
having better accuracy or, in case of a gas filled detector, improve on its accuracy by
using a more efficient gas mixture. Similarly easy steps can be taken to decrease the
systematic uncertainties associated with readout electronics. An obvious example of
that would be the use of an ADC having better resolution. Another way to decrease
the systematic uncertainty is to properly calibrate the system.
Systematic uncertainties are system specific and therefore there is no general
formula that could be used for their characterization. It is up to the experimenter
to carefully determine these errors and faithfully report them in the final results.
9.5.B RandomErrors..........................
Random errors refer to the errors that are statistical in nature. For example, the
radioactive decay is a random process. Even though we know theaveragerate of
decay of a sample, we can not predict when the next decay will happen. This implies
that there is aninherenttime uncertainty associated with the process. Similarly the
production of charge pairs in a radiation detector by passing radiation is also a
random process (see chapter 2). We can say thaton the averagehow many charge
pairs will be produced by a certain amount of deposited energy but we can not
associate an absolute number to it. Such uncertainties that are inherent to the
process and are statistical in nature are categorized as random uncertainties.
Fortunately most physical processes are Poisson in nature. This makes is fairly
easy to estimate the random error associated with a measurement. The random
error associated with a measurement during whichNcounts were recorded, is given
by
δstat=
√
N. (9.5.1)
For example, let us suppose that we measure the activity of a radioactive sample
by taking three consecutive readings by a single channel analyzer/counter: 2452,
2367, 2398. The absolute random errors associated with these measurements will
be:
δstat, 1 =√
2452 = 49. 52
δstat, 2 =√
2367 = 48. 65
δstat, 3 =√
2398 = 48. 97
9.5.C Error Propagation
Let us suppose we perform an experiment and makeNindependent measurements
xieach having uncertaintyδxiand standard deviationsigmaxi. We then use these
measurements to evaluate some functionu=f(x 1 ,x 2 , ..., xN). The question is: how
can we estimate the standard deviation and error in the quantity we thus determine?
This is where the error propagation formulae come into play, according to which the
combined variance and standard error in the functionucan be evaluated from
σ^2 u =[
∂f
∂x 1] 2
σ^2 x 1 +[
∂f
∂x 2] 2
σx^22 +....+[
∂f
∂xN] 2
σ^2 xN (9.5.2)and δu =[[
∂f
∂x 1] 2
δx^21 +[
∂f
∂x 2] 2
δx^22 +....+[
∂f
∂xN] 2
δx^2 N