Physics and Engineering of Radiation Detection

552 Chapter 9. Essential Statistics for Data Analysis

cumulative distribution function

p ≡ 1 −P(to)=1−

∫to

−∞

g(t|h)dt or

p =

∫∞

to

g(t|h)dt (9.6.1)

A single value ofp, however, does not tell us much about the agreement between
data and hypothesis because at each data point the level of agreement could be
different. The trick is to see how the value ofpis distributed throughout its range,
that is, between 0 and 1. Of course if there is perfect agreement, the distribution
will be uniform.
Let us now take a look at some of the commonly used goodness-of-fit statistics.

9.6.A Chi-Square (χ^2 )Test.......................

This is perhaps one of the most widely used goodness-of-fit statistic. In the following
we outline the steps needed to perform the test.

1.The foremost thing to do is to construct a hypothesis, which has to be tested.

This hypothesis should include a set of valuesμithat weexpectto get if we

perform measurements and obtain the valuesui. These set of values may have

been derived from a known distribution that the system is supposed to follow.

2.Decide on the number of degrees of freedom. If we takeNmeasurements, the

degrees of freedom are not necessarily equal toNbecause there may be one

or more relations connecting the measured valuesui.Ifthenumberofsuch

relations arer, then the degrees of freedom will be given byν=N−r.

3.Using the measured valuesui, compute a sample value ofχ^2 from the relation

9.3.53

χ^2 =

∑N

i=1

(ui−μi)^2

σi^2

.

4.Compute thenormalizedχ^2 ,thatis,χ^2 /ν.

5.Decide on the acceptable significance levelp, which represents the probability

that the data is in agreement with the hypothesis or not. A commonly chosen

value ofpis 0.05, which gives a confidence of 95%.

6.Determine the value ofχ^2 ν,pat whichpis equal to the chosen value. This means

evaluating the integral

p=

∫∞

χ^2 ν,p

f(x)dx (9.6.2)

forχ^2 ν,p(see also equation 9.6.1). f(x)isofcoursetheχ^2 probability den-

sity function. The solution to this equation requires numerical manipulations,

which can be done, for example by employing the Monte Carlo integration

technique. However this is not generally done since there are tables and graphs

available that can be used to deduce the values ofχ^2 ν,pwith respect topandν.