CCJ3300 Statistics In Criminal Justice Paper There is a minimum word requirement is a total of 1,200 words. The response should be supported with a minimum

CCJ3300 Statistics In Criminal Justice Paper There is a minimum word requirement is a total of 1,200 words. The response should be supported with a minimum of three scholarly references as well as in-text citations, using proper APA format. Please ensure you always cite any sentence that contains information from a source. (The references doesn’t count towards the word count!)

(All of the questions refer to the results of the SPSS analysis presented on pages 161-165 of the textbook.)

1. For the variable HOME, what are the modes? Is the data normally distributed?

2. For the variable ARREST, what are the modes? Is the data normally distributed?

3. Why are we concerned about the distribution of data?

4. What difference does it make in the case of each of the variables (HOME and ARREST) if the data is not normally distributed? © Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.
What do you
want to do?
Make
inferences
Describe
How many
variables?
Univariate
Bivariate
What level
of data?
Nominal
Ordinal
Central
tendency
Central
tendency
Mode
Dispersion
Dispersion
Range,
Index of
dispersion
Form
Form
Skew
Kurtosis
16304_CH06_Walker.indd 146
Kurtosis
M
I
L
E
S
,
S
H
A
N
N
O
NMedian
1
9Average
Absolute
0deviation
9
T
S Skew
Multivariate
Interval/
Ratio
Central
tendency
Mean
Dispersion
Variance,
Standard
deviation
Form
Skew
Kurtosis
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Chapter 6
The Form of a Distribution
Learning Objectives
Understand the number of modes,M
skewness, and kurtosis as they relate to
­explaining a data set.
I
?? Explain the difference between the mode and the number of modes.
L
?? Interpret the values of skewness and kurtosis as they relate to univariate
E
­analysis.
S curve in statistics.
?? Discuss the importance of the normal
?? Describe the properties of the normal
, curve.
??
The final univariate descriptive statistic, the form of the distribution, ties together the
central tendency and dispersion of the data.
S Three characteristics make up the form of
the distribution: the number of modes, the symmetry, and the kurtosis. In addressing
H
the form of a distribution, a polygon can generally be used to represent these characA
teristics visually.
N
N
6-1 Moments of a Distribution
In some statistics books and other places,O
distributions and the form of distributions are
referred to in terms of the moments of theNdistribution. There are four moments that are
considered important to a distribution. Moments are calculated as follows:
12 X2i
S1X
9N
where 1 X 2 X 2 represents the deviations0from the mean (as has been the case in Chapters 4 and 5), N is the total number of cases
9 in the distribution, and i is the moment
being calculated.
T the mean is always zero, the first moment
Since the sum of the deviations around
is always zero. If X is taken to the second
S power in the formula above, you can see
that this is the formula for the variance—thus, the second moment is the variance.
The third moment is usually associated with the skew of the distribution, although
the exact formula is to divide the formula for the third moment by the variance to
the power of 1.5. Similarly, the kurtosis of a distribution is associated with the fourth
147
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148??? Chapter 6
n
The Form of a Distribution
moment, although the exact formula is to divide the formula for the fourth moment by
the variance squared. The mean and variance were discussed in “Measures of Central
Tendency” and “Measures of Dispersion” (Chapters 4 and 5). The skew and kurtosis
of a distribution are discussed in this chapter, together with the third measure of the
form of a distribution: the number of modes.
6-2
Number of Modes
The first measure of the form of a distributionMis the number of modes. The number
of modes is important to higher-order analyses because it is indicative of the normalI
ity of the distribution. To use many bivariate and multivariate statistical procedures, a
L
unimodal distribution is preferred.
In determining the number of modes, aEslight deviation from determining the
mode may be necessary. Recall from the discussion of central tendency that it is comS
mon to count only the highest frequency in a distribution as the mode, even though
, should be considered. For determinsome people argue that all peaks of a distribution
ing the number of modes in an analysis of form, it may be more beneficial to look at
peaks rather than to find the one, highest value. Consider, for example, the distribution
S
in Figure 6-1. Even though there is only one highest value, there are three peaks in
H unsuitable for certain statistical prothe distribution. These peaks may make the data
cedures unless transformations are made. In this
A distribution, all three modes should
probably be counted in evaluating the form of the distribution even though the mode
N
is actually only 4.
N
O
N
45
40
1
9
0
9
T
S
35
30
25
20
15
10
5
0
1
2
3
4
Figure 6-1
16304_CH06_Walker.indd 148
5
6
7
8
9
10
Polymodal Distribution
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6-3
6-3
Skewness 149
Skewness
The next characteristic of the form of the distribution is the degree of symmetry
(skewness) of the distribution. This measure of the form of a distribution has three categories: symmetrical, positively skewed, and negatively skewed. A fully symmetrical
distribution has mirror-image sides such that the distribution could be split at the mean
and the sides folded over each other for a perfect match. In Figure 6-2, it is easy to see
the symmetry in the distribution. This is the histogram from Figure 4-7. The frequencies displayed in this distribution are very balanced: categories 1 and 7 have the same
M
frequency, as do 2 and 6, and 3 and 5. Category 4 has the highest frequency level. It
I be folded in half and the two sides would
is easy to see that this distribution could
match perfectly. This distribution is therefore,
L a perfectly symmetrical distribution. In
actual research, however, it is not common to see a perfectly symmetrical distribution.
E
More typically, the distribution will either be only close to symmetrical or not at all
S
symmetrical.
It should be noted here that the number
, of modes does not necessarily affect the
skew of the distribution. A distribution that is bimodal could still be cut in half where
the distribution mirrors itself. The only difference in this case is that the mode and
Snot be the same.
other measures of central tendency would
H
A
N
N
O
N
1
9
0
9
T
S
1
2
3
4
5
6
7
Figure 6-2 Histogram and Normal Curve for a Symmetrical Distribution
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150??? Chapter 6
n
The Form of a Distribution
Analysis of Skew
If a distribution is such that one side is different from the other, it is said to be skewed.
In skewed distributions, there is no point that can be drawn in the polygon where it could
be divided into two similar parts. If the point of the curve is to the left of the graph, it
is said to be negatively skewed (the tail of the graph points to the negative end of the
scale, smaller positive numbers). In Figure 6-3, “children” is an example of a negatively skewed distribution. Here, the point of the curve is toward category 1 or the left of
the graph. If the point of the curve is to the right of the graph, it is said to be positively
skewed (the tail of the graph points toward theM
positive end of the scale, larger positive
numbers). In Figure 6-3, “gun-wher” is an example
I of a positively skewed distribution.
Here, the point of the curve points toward category 12.5 or the right of the graph.
L output. A value of 0 means there
SPSS provides measures of skew in frequency
is no skew to the data. Skew values of zero are
E almost never obtained, however, and
a distribution is considered symmetrical if the skew value in SPSS is between 21 and
S
1.1 A distribution is generally considered skewed if it has a skew greater than +1.00
, The magnitude of the number will
or less than (a greater negative number) 21.00.
represent the degree of skew. When conducting research, it is desirable to obtain a
distribution that has a skew as close to zero as possible. If the skew is outside +1 to
S
21, the distribution may be too skewed to work with, and efforts should be made to
get the distribution closer to normal. This is H
done through transformations, which is
addressed in the discussion on regression. A
The frequency distribution that has been used with the other univariate measures
N
is shown in Table 6-1. Here the value of skew is 20.477, which means that the distribution is not perfectly symmetrical but thatNit exhibits an acceptable level of skew
(it is within the acceptable range of 0 to 21.00).
O There is some negative skew to this
distribution, as exhibited by the negative value, but it is not enough to warrant addiN
tional analyses or give cause for concern. If this value had been less than 21.00 (e.g.,
22.77), it might have been necessary to transform the distribution.
1
9
0
9
T
S
30
400
300
20
200
10
100
0
Std. Dev. = 0.22
Mean = 1.95
N = 324.00
1.00
CHILDREN
1.50
2.00
0
Std. Dev. = 1.85
Mean = 2.2
N = 47.00
0.0
2.5
5.0
7.5
10.0
12.5
GUN_WHER
Figure 6-3 Negatively and Positively Skewed Distributions
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6-3
Skewness 151
What is your highest level of education?
Value Label
Value
Frequency
Percent
Valid
Percent
Cumulative
Percent
Less than High School
1
16
?? 4.6
?? 4.8
?? 4.8
GED
2
59
17.0
17.6
22.3
High School Graduate
3
?? 8
?? 2.3
?? 2.4
24.7
Some College
4
117
33.7
34.8
59.5
College Graduate
5
21.4
81.0
6
M72
64
I 11
L
347
E
336
S11
, 4.08
20.7
Post Graduate
18.4
19.0
100.0
?? Missing
?? Total
Valid
N
Missing
Mean
Median
?? 3.2
100.0
100.00
4.00
Mode
Std. Deviation
Variance
Skewness
Std. Error of Skewness
Kurtosis
Std. Error of Kurtosis
Table 6-1
S4
H1.460
2.131
A2.477
N .133
N2.705
O .265
N of Measures of Form
SPSS Output
Although quantitative measures of 1
skewness and kurtosis will almost always be
available when conducting actual research, an estimate of the skew of a distribution
9
can be made even without a skew calculation. If the mean and the median are different,
0 although it is not possible to tell if it is
the distribution is at least somewhat skewed,
beyond +1 or 21. Additionally, the skew9is in the direction of the mean. For example,
if the distribution is positively skewed, the mean will be larger than the median, but
T
if the skew is negative, the mean will be smaller than the median. It should also be
S
noted that the mode is generally on the opposite
side of the median from the mean in
skewed distributions. This is not always the case, however, and should not be treated
as a rule. MacGillivray (1981) discusses the conditions under which each of these
examples would fall.
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152??? Chapter 6
6-4
n
The Form of a Distribution
Kurtosis
The last characteristic of the form of a distribution is the kurtosis. For kurtosis, think
again of stacking blocks (or beer cans) on top of each other to represent the frequency
of the categories in a histogram. The kurtosis is the extent to which cases are piled up
around the measure of central tendency or in the tails of the distribution. If most of the
values in the distribution are very close to the measure of central tendency, the distribution is said to be leptokurtic (as shown on the left in Figure 6-4). If most of the
values in the distribution are out in the tails, the distribution is said to be platykurtic,
M
as shown on the right in Figure 6-4. If the values in the distribution are such that they
represent a distribution such as that shown in IFigure 6-2, the distribution is said to be
mesokurtic, as shown in the center of FigureL
6-4. It is desirable to have a mesokurtic
distribution in research; otherwise, the data may have to be transformed.
E
S
,
1
S
H
A
1
2
3
4
7
2
3
4
5
6
N5 6 7 1 2 3 4 5 6 7
Figure 6-4 Leptokurtic, Mesokurtic,
Nand Platykurtic Distributions
O
The shape of these curves also offers anN
opportunity to talk about variance and
standard deviation. As discussed in “Measures of Dispersion” (Chapter 5), the variance and standard deviation dictate the shape of the distribution. In a leptokurtic distribution, the variance and standard deviation1would be smaller than in a mesokurtic
distribution. The variance and standard deviation
9 of a platykurtic distribution would
be larger than either a mesokurtic or leptokurtic distribution. This is one application of
0
the variance and standard deviation. A more expanded discussion of this application is
9
presented later in the section on the normal curve.
Analysis of Kurtosis
T
S
In SPSS, kurtosis is measured in the same way as skew. A value between +1 and 21
represents a mesokurtic distribution. Positive numbers greater than 1 represent leptokurtic curves. Negative numbers less than 21 (a greater negative number) represent
platykurtic curves. As with skew, it is desirable to get the kurtosis as close as possible
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6-6
Design of the Normal Curve 153
to zero, using transformations if necessary. Examining the kurtosis value in Table 6-1
shows that the distribution is mesokurtic because the value (20.705) is between 21.00
and 0. If this value had been 21.705, the distribution would have been platykurtic.
6-5
The Importance of Skew and Kurtosis
It is important to know the skew and kurtosis because some statistical procedures do not
work well with skewed data or data that is not mesokurtic. If data in a research project is
found to be skewed or kurtose, it may be necessary to transform the data. Initially, you
M
must remember two things about transformation. First, if the data is not within acceptable tolerances for skew and kurtosis, theIdata will need to be transformed prior to using
some statistical procedures. Second, after
L making transformations, recheck both the
skew and kurtosis. Transforming the data may bring one of these within acceptable tolE
erances but may make the other unacceptable. If that happens, you will need to choose
another transformation. You should thenSrecheck the skew and kurtosis again. This
process should continue until you reach ,a point where both the skew and kurtosis are
acceptable. If it is not possible to get both the skew and kurtosis in an acceptable range,
you may need to consider using a different analysis procedure that is not susceptible to
S
nonnormal curves.
H
6-6 Design of the Normal
A Curve
Extending the concepts of the frequencyN
distribution, graphical representation of data,
and measures of central tendency, dispersion,
N and form brings us to the point of discussing a key concept in statistical analysis, the normal curve. At this point, you
Oof the normal curve; that is covered in more
should not be concerned with applications
detail in the chapters of the book on inferential
analysis. The purpose of the present
N
discussion is to introduce the properties of the normal curve.
An introduction to the normal curve is included in descriptive analyses rather than
1 the normal curve can be used to provide
inferential analyses for two reasons. First,
an interpretation of the variance and standard
9 deviation. Second, the normal curve is
important to a number of statistical procedures that will be discussed before reaching
0
information on inferential statistical procedures.
An example of relatively normally 9
distributed data can be shown in grades in a
course (see Figure 6-5). Say that most people
T taking the course score a C on the first
test. This would be the modal grade (the top part of the curve). There are those who
S
receive high A’s, but there would be only a few of these; they would be at the positive
end of the curve. There are also those who receive very low F’s, but these are also few;
they would be at the negative end of the curve. Most people would be in between these
two extremes, with more people making scores around C’s than other grades, and
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154??? Chapter 6
n
The Form of a Distribution
Figure 6-5
M
I
L
E
S
Normal Curve
,
more people making B’s and D’s than A’s and F’s. For the sake of argument, though,
say that no one received a 100 and there were a few people who did not take the test.
S
Therefore, the ends of the tails will never completely touch the baseline.
This type of data represents a special formHof distribution called the normal curve.
This type of curve or distribution is very much
A like those that have been used in this
chapter and at the end of Chapter 4. A normal curve is special because it has certain
N
characteristics. First, a normal curve is symmetrical in that it can be folded in half,
and both sides would be exactly the same (asN
in Figure 6-2, where the frequencies of
category 1 and 3 are the same as those of category
O 5 and 7, respectively). Note, though,
that this does not mean this curve cannot be kurtose. Some symmetrical distributions
N
are leptokurtic or platykurtic. This is shown in Figure 6-4, where each of those distributions was symmetrical even if it was kurtose. Also, a normal curve is unimodal;
there is one, and only one, peak. This peak is1at the maximum frequency of the data
distribution, so that the mean, median, and mode all have the same value. The normal
9
curve shown in Figure 6-5 has only one mode. From the peak, the tails of a normal
0 always getting closer to the baseline
curve fall off on both ends and extend to infinity,
but never touching it. This is shown in Figure96-5, where the bottom part of the curve
straightens out and runs relatively parallel to the X axis. You may say this makes no
T
sense; all distributions have an end, so why would the normal curve not have an end?
The answer lies in the scientific process. TakeS
the example of computers. Less than 20
years ago, scientists and engineers thought they had achieved the ultimate when they
were able to reach 640K of random access memory (RAM) in a computer. They felt
that this was the maximum that could be achieved and all that anyone would ever need.
To them, the distribution limits were set. We now know, of course, that 640K was only
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6-6
Design of the Normal Curve 155
the beginning and that computers are far beyond that now. It would have been foolish,
then, to have the curve touch the line at 640K; it should not touch the line because we
do not know what will come in the future. A final characteristic of the normal curve
that merits discussion is that the area under a normal curve is always the same, regardless of the data set. The area under the normal curve is 1.00, or 100% of all values in
the distribution. This is extremely important in the section of this book concerning
inferential analyses because of its importance in estimating the placement of a sample
distribution within a population or another sample.
The area under the normal curve also
M offers the opportunity to put the variance
and standard deviation into practice. Say, for example, that a researcher was examinI before they committed another crime or
ing the time prisoners were out on parole
L If the time each parolee took before being
returned to prison on a technical violation.
reincarcerated was plotted, it might lookEas in Figure 6-6. There were a few people
who returned to prison right away, most of the parolees who returned to prison did it
within two to four years, and some tookS
longer. Some had not recidivated at the time
of the research, so the end of the distribution
, is open.
An analysis of the central tendency would put the mean of this distribution at 36
months, which is represented by the vertical line. This is good information: the averS
age length of time for parolees to be reincarcerated
is three years. It is obvious from
this distribution, however, that not all ofH
the parolees were reincarcerated at the same
time. The span of time runs from a couple o…
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