Page 68 - Computational Statistics Handbook with MATLAB

54 Computational Statistics Handbook with MATLAB

n
1
X = --- ∑ X i . (3.1)
n
i = 1
To calculate this in MATLAB, one can use the function called mean. If the
argument to this function is a matrix, then it provides a vector of means, each
one corresponding to the mean of a column. One can find the mean along any
dimension (dim) of multi-dimensional arrays using the syntax:
mean(x,dim).
Another statistic that we will see again is the sample variance, calculated
from

n  n  n  
2
1
1
2
2
2

S = ------------ ∑ ( X i – X) = -------------------- n ∑ X i –  ∑ X i .  (3.2)
(
n – 1 nn – 1)    
i = 1 i = 1 i = 1
The sample standard deviation is given by the square root of the variance
(Equation 3.2) and is denoted by . These statistics can be calculated in
S
MATLAB using the functions std(x)and var(x), where x is an array con-
taining the sample values. As with the function mean, these can have matri-
ces or multi-dimensional arrays as input arguments.
Sample
oments
SampleM M oments
SSampleample
MM omentsoments
The sample moments can be used to estimate the population moments
described in Chapter 2. The r-th sample moment about zero is given by
n
1 r
M' r = --- ∑ X i . (3.3)
n
i = 1
Note that the sample mean is obtained when r = 1 . The r-th sample
moments about the sample mean are statistics that estimate the population
central moments and can be found using the following

n
1 r
M r = --- ∑ ( X i – X) . (3.4)
n
i = 1
We can use Equation 3.4 to obtain estimates for the coefficient of skewness
. Recall that these are given by
γ 1 and the coefficient of kurtosis γ 2

© 2002 by Chapman & Hall/CRC