2.4 The Chi-Square Distribution
As we will see in Section 7, the chi-square
distribution is
particularly useful for testing the goodness-of-fit of theoretical
formulae to experimental data. Mathematically, the chi-square is
defined in the following manner. Suppose we have a set of n
independent random variables, xi, distributed as
Gaussian densities
with theoretical means ยตi and standard
deviations 
i,
respectively. The sum
is then known as the chi-square. This is more often designated by the
Greek letter 
2; however, to avoid confusion due to the
exponent we
will use u = 2 instead. Since xi is a
random variable, u is also a
random variable and it can be shown to follow the distribution
where v is an integer and
(v / 2) is the gamma
function. The integer v
is known as the degrees of freedom and is the sole parameter of the
distribution. Its value thus determines the form of the distribution.
The degrees of freedom can be interpreted as a parameter related to
the number of independent variables in the sum (22).
|
Fig. 6. The chi-square distribution for various values of the degree of freedom parameter v. |
Figure 6 plots the chi-square distribution for various values of v. The mean and variance of (23) can also be shown to be
To see what the chi-square represents, let us examine (22) more closely. Ignoring the exponent for a moment, each term in the sum is just the deviation of xi from its theoretical mean divided by its expected dispersion. The chi-square thus characterizes the fluctuations in the data xi. If indeed the xi are distributed as Gaussians with the parameters indicated, then on the average, each ratio should be about 1 and the chi-square, u = v. For any given set of xi, of course, there will be a fluctuation of u from this mean with a probability given by (23). The utility of this distribution is that it can be used to test hypotheses. By forming the chi-square between measured data and an assumed theoretical mean, a measure of the reasonableness of the fluctuations in the measured data about this hypothetical mean can be obtained. If an improbable chi-square value is obtained, one must then begin questioning the theoretical parameters used.