ch02 JWBK035-Vince February 12, 2007 6:50 Char Count= 0
Probability Distributions 47The final measure of central tendency is thequadratic meanorroot
mean square.R^2 = 1 /N∑N
i= 1X^2 i (2.04)where: R=The root mean square.
Xi=The ith data point.
N=The total number of data points in the distribution.
You should realize that the arithmetic mean (A) is always greater than
or equal to the geometric mean (G), and the geometric mean is always
greater than or equal to the harmonic mean (H):H<=G<=A (2.05)where: H=The harmonic mean.
G=The geometric mean.
G=The arithmetic mean.Moments of a Distribution
The central value or location of a distribution is often the first thing you
want to know about a group of data, and often the next thing you want
to know is the data’s variability or “width” around that central value. We
call the measures of a distribution’s central tendency thefirst momentof a
distribution. The variability of the data points around this central tendency
is called thesecond momentof a distribution. Hence, the second moment
measures a distribution’s dispersion about the first moment.
As with the measure of central tendency, many measures of dispersion
are available. We cover seven of them here, starting with the least common
measures and ending with the most common.
Therangeof a distribution is simply the difference between the largest
and smallest values in a distribution. Likewise, the10–90 percentile range
is the difference between the 90th and 10th percentile points. These first
two measures of dispersion measure the spread from one extreme to the
other. The remaining five measures of dispersion measure the departure
from the central tendency (and hence measure the half-spread).
Thesemi-interquartile range or quartile deviationequals one half of
the distance between the first and third quartiles (the 25th and 75th per-
centiles). This is similar to the 10–90 percentile range, except that with this
measure the range is commonly divided by 2.