Physics and Engineering of Radiation Detection

532 Chapter 9. Essential Statistics for Data Analysis

across situations where the data can not be represented by a distribution for

which the integral ofg(x)nf(x) can be easily evaluated to find the moments.

There is way out of this situation, however. Even though the moments are not

calculable for some functions but their Fourier transforms can be determined

by evaluating the integral

ψ(ν)=

∫∞

−∞

eiνxf(x)dx. (9.3.12)

Whereiis the complex number andνis a parameter having dimensions that

are inverse of the parameterx. For example, ifxrepresents time thenνis

the frequency. This transformation of the distribution function dependent on

xinto acharacteristic functiondependent onνis equivalent to taking the

expectation value ofeiνx. The good thing about this characteristic function is

that it can be used to determine the moments off(x) through the relation

αn=i−n

dnψ

dνn

∣

∣

∣

ν=0

. (9.3.13)

Skewness:Not all real distribution functions are symmetric. In fact, we sel-

dom find a parameter that can be represented by a distribution function hav-

ing no skewness whatsoever (see Fig.9.3.1). Unless this skewness is negligibly

small, it must be quantified to extract useful information from the distribu-

tion. The best and the most commonly used way for this quantification is the

computation of the so calledcoefficient of skewnessdefined as

γ 1 =

m 3

σ^3

. (9.3.14)

The higher the value ofγ 1 , the more skewed is the distribution. Althoughγ 1

is extensively used in analyses, but in fact any odd moment about the mean

can be used as a measure of skewness.

Kurtosis: Besides skewness, the tail of a distribution also contains useful

information and should not be neglected in the analysis. A commonly used

measure of this tail is known as kurtosis of the distribution and is defined as

γ 2 =

m 4

σ^4

− 3. (9.3.15)

The reason for defining kurtosis in this way lies in the use of the Gaussian

distribution as a standard for comparison. As we will see later in this chapter,

most of the natural processes can be described by the Gaussian distribution

and therefore using it as a standard is justified. A perfect Gaussian distribution

has a kurtosis of 0 since it always satisfiesm 4 =3σ^4 .Notethatγ 2 =0means

that the tail of the distributionfalls offaccording to the Gaussian distribution,

not that it does not have any tail. If the distribution has a long tail, that

is, if kurtosis is negative, it is called aplatykurticdistribution. Aleptokurtic

distribution, on the other hand, has positiveγ 2 and its tail falls off quicker

than the Gaussian distribution.