Physics and Engineering of Radiation Detection

566 Chapter 9. Essential Statistics for Data Analysis

This statement needs some clarification, though. What we are assuming here is that
the measurement time is not changing. If we increase the measurement time,N
would increase and so would the measurement precision. Hence the physical limit
on measurement precision actually depends on the measurement time.
The discussion above applies to measurement at a point only. How can we de-
termine the precision in measurement if measurements are made at several points?
An obvious example is energy spectroscopy, where the goal is to obtain an energy
spectrum of particles. Such measurements generally produce one or more peaks over
a background. The main analysis tasks are to identify the peaks and measure their
respective areas. Both of these tasks are tied to the identification and elimination of
background. The difficulty lies in determining the true area of the peak, that is the
counts that contributed to the peak and not the background. To simplify the mat-
ter, let us first assume that the peak is background-free as shown in Fig.9.11.1. This
is a Gaussian-like peak with two tails. Now, the tails are not part of the background
but could have arisen because of the errors induced by the measuring device^1 .That
is, there is some uncertainty associated with them. Hence we would want to exclude
them from the measurement of peak precision. This requires selection of aregion of
interestin the peak as shown in the figure. It should be noted that there is no uni-
versally accepted method of selecting the region of interest but most experimenters
use the area above a line that cuts the peak at 10% of its maximum. The width of
the peak at this line is calledfull width at one tenth of the maximum.

FWTM

1/10 of Max.

Region of

Interest

FWHM

Figure 9.11.1: Distribution of

counts in an experiment with

no background. The region of

interest (shaded portion), with

an area ofNi, has been chosen

such that the counts below 10%

of the peak amplitude get dis-

carded. The total area of the

peak isNt.

It is apparent that the best error estimate will be obtained if we take all the counts
in the peakNt,thatis

δNt=

1

√

Nt

. (9.11.4)

(^1) Some experimenters prefer to include tails in the background as well. However a better approach is to
treat tails and background separately.