Geoscience Reference
In-Depth Information
a
b
Fig. 3.10 a
Probability density function
f
(
x
), and
b
cumulative distribution function
F
(
x
), of a
Fisher's
F
distribution with dif erent values
for ʦ1 and ʦ2.
of freedom. In the analysis of univariate data this distribution has
n
-1
degrees of freedom, where
n
is the sample size. As ʦ→∞, the
t
distribution
converges towards the standard normal distribution. Since the
t
distribution
approaches the normal distribution for ʦ>30, it is rarely used for distribution
i tting. However, the
t
distribution is used for hypothesis testing using the
t
-test (Section 3.7).
Fisher's
F
Distribution
h e
F distribution
was named at er the statistician Sir Ronald Fisher (1890-
1962). It is used for hypothesis testing using the
F
-test (Section 3.8). h e
F
distribution has a relatively complex probability density function (Fig. 3.10):
where
x
>0 and ʓ is again the Gamma function. h e two parameters ʦ
1
and
ʦ
2
are the numbers of degrees of freedom.
χ
2
or Chi-Squared Distribution
h e ˇ
2
distribution was introduced by Friedrich Helmert (1876) and Karl
Pearson (1900). It is not used for i tting a distribution but has important
applications in statistical hypothesis testing using the ˇ
2
-test (Section 3.9).
