Physics and Engineering of Radiation Detection

548 Chapter 9. Essential Statistics for Data Analysis

with respect toσ,welookatsometypicalvalues.

P(μ−σ<x<μ+σ)=0. 6827

P(μ− 2 σ<x<μ+2σ)=0. 9545

P(μ− 3 σ<x<μ+3σ)=0. 9973

What these values essentially show is that if the data can be represented by
a perfect Gaussian distribution, then we can be only 68.27% sure that the next
measurement will lie within the rangeμ±σ. However if we wanted to be more than
99% sure about this we will have to stretch the range to around 3σon both sides of
the distribution. Fig.9.4.2 explains this concept in graphical form.

f(x)

2 σ

x

Figure 9.4.2: Confidence interval of a stan-

dard Gaussian distribution. The shaded area

represents the probability that the next mea-

surement ofxwill lie within the intervalμ−σ<

x<μ+σ. For a perfect Gaussian distribution

this turns out to be 0.6827 meaning that one

could be up to 68.27% sure that the value will

not be out of these bounds.

9.5 MeasurementUncertainty

There is always some uncertainty associated with a measurement no matter how
good our measuring device is and how carefully we perform the experiment. There
are different types of uncertainties associated with any measurement but they can
be broadly divided into two categories: systematic and random.

9.5.A SystematicErrors

All measurements, direct or indirect, are done through some type of measuring de-
vice. Since there is no such thing as a perfect device, therefore one should expect
some error associated with the measurement. This type of error falls into the cate-
gory of systematic errors, which refer to the uncertainties in the measurement due
to the measurement procedures and devices. Unfortunately it is not always easy
to characterize systematic errors. Repeating the measurements does not have any
effect on them since they are not random. In other words, systematic errors are not
statistical in nature and therefore can not be determined by statistical methods.
The good thing is that the systematic uncertainties can be minimized by modi-
fying the procedures and using better devices. For example, one can use a detector