Physics and Engineering of Radiation Detection

570 Chapter 9. Essential Statistics for Data Analysis

standard deviation in measurement ofNiis given by

σi=

[

σ^2 t+σb^2

] 1 / 2

, (9.11.24)

where the subscriptsi,t,andbrefer to the peak, the total, and the background
respectively. The spread in the total counts is given byσt=

√

Nt. To calculate the
spread in the background counts we note from equation 9.11.22 that sincevandw
are constants, the spread inNbis equal to the spread inNb 1 +Nb 2 ,thatis

σb≈

w

v

[Nb 1 +Nb 2 ]^1 /^2. (9.11.25)

This can be simplified by substituting the value ofNb 1 +Nb 2 from equation 9.11.22.
Hence we get

σb≈

w

v

[v

w

Nb

] 1 / 2

. (9.11.26)

Substitution ofσtandσbin equation 9.11.25 yields

σi=

[

Nt+

w

v

Nb

] 1 / 2

. (9.11.27)

This equation can also be written in terms ofNiby substitution ofNt=Ni+Nb.

σi =

[

Ni+Nb+

w

v

Nb

] 1 / 2

=

[

Ni+

(

1+

w

v

)

Nb

] 1 / 2

(9.11.28)

Dividing both sides of this equation byNigives us the error in measurement ofNi.

σi

Ni

≡δNi=

1

√

Ni

[

1+

(

1+

w

v

)N

b

Ni

] 1 / 2

(9.11.29)

Note that, in the absence of background (Nb= 0), this equation reduces to equation
9.11.5. This equation sets the limit to accuracy of the measurement of peak area
using our simple background elimination technique. To get an idea of how it com-
pares to the minimum possible errorδNt=1/

√

Nt, let us divide it on both sides by
δNt.

δNi

δNt

=

√

Nt

Ni

[

1+

(

1+

w

v

)Nb

Ni

] 1 / 2

(9.11.30)

One conclusion we can draw here is that the error in peak area depends on the total
background counts. This is due to the statistical fluctuations in the background
counts. So, the common notion thatthe size of the background does not matter for
as long as it is constant, is not really correct. The only way background can be
suppressed is by improving the experimental setup. For example, proper shielding
can suppress background considerably.