Page 85 - Computational Statistics Handbook with MATLAB

Chapter 3: Sampling Concepts 71

q
0.5
0.5 = ∫ fx()d . x (3.42)
– ∞

We can get a measure of the dispersion of the random variable by looking at
the interquartile range (IQR) given by

IQR = q 0.75 – q 0.25 . (3.43)

One way to obtain an estimate of the quantiles is based on the empirical
, , ,
distribution function. If we let X 1() X 2() … X n() denote the order statistics for
is an estimate of the j –( ⁄
a random sample of size n, then X j() 0.5) n quantile
[Banks, 2001; Cleveland, 1993]:

1 j –
0.5
X j() ≈ F –   ---------------   . (3.44)
n
We are not limited to a value of 0.5 in Equation 3.44. In general, we can esti-
mate the p-th quantile using the following

j
,
---------- <
,
ˆ
q p = X ; j – 1 p ≤ ---; j = 1 … n . (3.45)
j ()
n n
As already stated, Equation 3.45 is not the only way to estimate quantiles.
For more information on other methods, see Kotz and Johnson [Vol. 7, 1986].
The analyst should exercise caution when calculating quartiles (or other
quantiles) using computer packages. Statistical software packages define
them differently [Frigge, Hoaglin, and Iglewicz, 1989], so these statistics
might vary depending on the formulas that are used.

EXAMPLE 3.5
In this example, we will show one way to determine the sample quartiles.
ˆ
The second sample quartile q 0.5 is the sample median of the data set. We can
calculate this using the function median. We could calculate the first quartile
ˆ
q 0.25 as the median of the ordered data that are at the median or below. The
ˆ
third quartile q 0.75 would be calculated as the median of the data that are at
ˆ
q 0.5 or above. The following MATLAB code illustrates these concepts.
% Generate the random sample and sort.
x = sort(rand(1,100));
% Find the median of the lower half - first quartile.
q1 = median(x(1:50));
% Find the median.
q2 = median(x);

© 2002 by Chapman & Hall/CRC