Investigations of Faraday Rotation Maps of Extended Radio Sources ...

1.4. SOME STATISTICAL TOOLS 23
and
σ 2 ϕ 0
=
λ 4
. (1.59)
S λ 4 − λ22 Another quantity worth to consider is the covariance defined as
cov(y i , y j ) = 〈(y i − 〈y i 〉)(y j − 〈y j 〉)〉 = 〈y i y j 〉 − 〈y i 〉 〈y j 〉 , (1.60)
where the brackets denote the expectation value. The variance is defined as
var(y i ) = cov(y i , y i ). The square root of the variance is the standard deviation σ i
in a measurement. However, one can understand the covariance cov(y i , y j ) as elements
of a covariance matrix which describes correlations among the y i ’s. If the y i
and y j were totally uncorrelated the off-diagonal elements of the covariance matrix
would be zero.
Another interesting point related to the interpretation of measurements is the optimal
error weighting of any data set. Consider a data set x i , x j , ..., x n with given
standard errors σ 1 , σ 2 , ..., σ n . Then any error weighted mean x can be expressed as
x =
∑ ni=1
x i /σ α i
∑ ni=1
1/σ α i
, (1.61)
where the exponent α is a parameter for which the above expression has to be optimised.
Following Gaussian error propagation, the standard error of the mean would
be
n∑
( )
σ 2 ∂x 2
x = σ i . (1.62)
∂x i
The evaluation of the derivation in Eq. (1.62) using Eq. (1.61) leads to
σ x 2 =
i=1
∑ ni=1
σ 2−2α
i
( ∑ n
i=1 1/σ i α ) 2 (1.63)
The best data weighting is achieved when the resulting error is the smallest. Therefore,
the minimum with respect to α of Eq.(1.63) is required
⎛
0 = ∂σ x
∂α = ⎝
n∑
i=1
ln σ i σ 2−2α
i
n∑
j=1
σ −α
j
⎞ ⎛
n∑
⎠ − ⎝
i=1
σ 2−2α
i
⎞
n∑
ln σ j σj
−α
j=1
⎠ . (1.64)
For this equation to be true the condition 2 − 2α = −α has to be fulfilled. Therefore,
the optimal error weighting for the calculation of the mean of a data set is achieved for
α = 2.
1.4.2 Autocorrelation and Power Spectra
As discussed in the previous section, the magnetic field in clusters of galaxies seems
to consist of a large-scale component and of a fluctuating component (see Sect. 1.3).
This fluctuating component can be statistically described by using the power spectrum
and the autocorrelation function. The concept of these descriptive statistics and their
properties is introduced in the following (for a review see Kaiser 2003).