 Various
Measures of Dispersion
In many ways, measures of
central tendency are less useful in statistical
analysis than measures of dispersion of values around
the central tendency
 The dispersion of values
within variables is especially important in social and
political research because:
 Dispersion or
"variation" in observations is what we seek to
explain.
 Researchers want to know
WHY some cases lie above average and others below average
for a given variable:
 TURNOUT in voting:
why do some states show higher rates than
others?
 CRIMES in cities: why
are there differences in crime rates?
 CIVIL STRIFE among
countries: what accounts for differing
amounts?
 Much of statistical
explanation aims at explaining DIFFERENCES in
observations  also known as
 VARIATION, or the
more technical term, VARIANCE.

 The SPSS Guide
contains only the briefest discussion of measures of
dispersion on pages 2324.
 It mentions the
minimum and maximum values as the extremes,
and
 it refers to the
standard deviation as the "most commonly used"
measure of dispersion.
This is not enough, and
we'll discuss several statistics used to measure variation,
which differ in their importance.
 We'll proceed from
the less important to the more important,
and
 we'll relate the
various measures to measurement
theory.

EasytoUnderstand Measures of dispersion for NOMINAL
and ORDINAL variables
In the great scheme of
things, measuring dispersion among norminal or oridinal
variables is not very important.
 There is inconsistency
in methods to measure dispersion for these variables,
especially for nominal variables.
 Measures suitable for
nominal variables (discrete, nonorderable) would also
apply to discrete orderable or continuous variables,
orderable, but better alternatives are
available.
 Whenever possible,
researchers try to reconceptualize nominal and ordinal
variables and operationalize (measure) them with an
interval scale.
 Variation Ratio,
VR
 VR = l  (proportion
of cases in the mode)
 The value of VR
reflects the following logic:
 The larger the
proportion of cases in the mode of a nominal variable,
the less the variation among the cases of that
variable.
 By subtracting the
proportion of cases from 1, VR reports the
dispersion among cases.
 This measure has
an absolute lower value of 0, indicating NO
variation in the data (occurs when all the cases
fall into one category; hence no
variation).
 Its maximum value
approaches one as the proportion of cases inside
the mode decreases.
 Unfortunately, this
measure is a "terminal statistic":
 VR does not
figure prominently in any subsequent procedures for
statistical analysis.
 Nevertheless, you
should learn it, for it illustrates
 that even nominal
variables can demonstrate variation
 that the variation
can be measured, even if somewhat
awkwardly.

Easytounderstand measures of variation for
CONTINUOUS variables.

 RANGE: the
distance between the highest and lowest values in a
distribution
 Uses information on
only the extreme values.
 Highly unstable as a
result.
 SEMIINTERQUARTILE
RANGE: distance between scores at the 25th and the
75th percentiles.
 Also uses information
on only two values, but not ones at the
extremes.
 More stable than the
range but of limited utility.
 AVERAGE
DEVIATION:

where
= absolute value of the
differences

 Absolute values
of the differences are summed, rather than the
differences themselves, for summing the positive and
negative values of differences in a distribution
calculating from its mean always yields 0.
 The average deviation
is simple to calculate and easily
understood.
 But it is of limited
value in statistics, for it does not figure in
subsequent statistical analysis.
 For mathematical
reasons, statistical procedures are based on measures
of dispersion that use SQUARED deviations from the
mean rather than absolute
deviations.

Hardtounderstand measures of variation for
CONTINUOUS variables.
SUM OF SQUARES is the sum of the
squared deviations of observations from the
mean of the distribution.
 The SUM OF SQUARES
is commonly represented as SS
 The formula is
 This quantity is also
known as the VARIATION.
 It appears in the
numerator of formulas for standard deviation and
variance (below).
 We will see later
that this value is useful on its own.
 Here is a
"worksheet" for comuting the SUM OF
SQUARES:
Values of
X_{i}
(1)

Deviations of X_{i
}from the mean
(2)

Deviations squared
(3)





5

0

0

9

4

16

2

3

9

8

3

9

6

1

1

5

0

0

4

1

1

7

2

4

4

1

1

3

2

4

1

4

16

6

1

1






Sum of the Squared
Deviations

VARIANCE is simply
the mean of these squared and summed deviations (i.e.,
the average of the squared deviations)
 The symbol for
VARIANCE is s^{2} accompanied by
a subscript for the corresponding
variable.
 Here's the formula
for variance of variable X:
 For the data above,
the variance computes as
STANDARD
DEVIATION, is the square root of the
variance
 The symbol for
STANDARD DEVIATION (often shortened to STD
DEV) is s
 The formula for the
standard deviation of variable X:
 For the data above,
the standard deviation computes as
 This is the
most important formula in statistics
 But read on page
24 of the SPSS Guide that "for theoretical
reasons" (n1) is used in the demoninator to
compute the standard devation, not n.
 Indeed, SPSS
routinely uses n1 and not n when it
computes the standard deviation.
 But for now, learn
this formula as the definition of the standard
deviation.
 We will get to the
"theoretical reasons" later.


Dispersion
and the Normal Distribution
Occasionally in lecture I spoke about the "shape"
of a distribution of values, using these terms:
 unimodal  the distribution had only a single
value that occurred most frequently
 symmetrical  the left side of the
distribution of values mirrored the right side, i.e, it
was neither
 skewed to the left, nor was it
 skewed to the right
 bellshaped  the frequencies of cases
declined toward the extreme values in the right and left
tails, so that the distribution had the appearance of a
"bell."
When such a "bellshaped" distribution demonstrates
certain properties, it is said to have a normal
distribution.
 In one sense, it is
called "normal" because a unimodal, symmetrical,
bellshaped distribution "normally" develops in the long
run through accidents of nature when the events are
equallylikely to occur.
 In an infinite amount
of time, a random process could ultimately generate
any structured result: e.g., a group of monkeys seated
at typewriters could peck out all the great works of
literature.
 This would be an
extremely rare event, but it is
conceivable.
 The normal curve is a
mathematical formula that assigns probabilities to the
occurrence of rare events.
 Statistically speaking,
it is a probability distribution for a continuous random
variable:
 The ordinate
represents the probability density for the occurrence
of a value.
 The baseline
represents the values.
 The exact shape of
the curve is given by a complicated formula that you
do NOT need to know.
 The area under the
curve is interpreted as representing all occurrences
of the variable, X.
 We can consider
the area as representing 100 PERCENT of the
occurrences; in PROPORTIONS this is expressed as
1.0.
 We can then
interpret AREAS under the curve as representing
certain PROPORTIONS of occurrences or
"probabilities".
 We cannot assign a
probability to any point, but we can attach
probabilities to INTERVALS on the baseline
associated with AREAS under the curve: e.g., the
mean has 50% of the cases standing to each
side.

Special properties of the normal
distribution:
 Its shape is such that
it
 Embraces
68.26% of the cases within 1 s.d. around
the mean.
 Embraces
95.46% of the cases within 2 s.d. around
the mean.
 Embraces
99.74% of the cases within 3 s.d. around
the mean.
 More roughly speaking,
68%, 95%, and 99% of the cases are embraced
within 1, 2, and 3 standard deviations from
the mean in a normal distribution.
Determining whether a distribution is
"normal"
Assignment for today: Compute measures of dispersion
for pctwomen, pctblack, and bush2000.
Which is closest to a "normal distribution"?
