QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT 1
Marcos Reis Campos, MBA
BUSI 820
Liberty University
May 26, 2024
Table of Contents
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT 2
Chapter 3 – SPSS Problems.............................................................................................................4
3.1. Bar chart: Why did/did not create one for each variable..........................................................4
Bar Chart Discussion.......................................................................................................................4
3.2. Histograms: Why did/did not create one for each variable.......................................................5
Histogram Discussion......................................................................................................................5
3.3 (a). Frequency polygons: Why did/did not create one for each variable...................................6
Frequency Polygons Discussion......................................................................................................6
3.4. Range, standard deviation, skewness: which are meaningful...................................................6
Range...............................................................................................................................................6
Standard Deviation...........................................................................................................................7
Skewness..........................................................................................................................................7
Range, Standard Deviation, Skewness Discussion..........................................................................7
3.5. Mean, median, mode: which are meaningful............................................................................8
Mean................................................................................................................................................8
Median.............................................................................................................................................8
Mode................................................................................................................................................8
Mean, Median, Mode Discussion....................................................................................................8
Figures............................................................................................................................................10
Figure 1: Mother’s Education........................................................................................................11
Figure 2: Math Achievement
Test..................................................................................................12
Histograms……………………………………………………………………………………….13
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT 3
Figure 3:
Ethnicity..........................................................................................................................13
Figure 4: Academic Tract..............................................................................................................14
Figure 5: Math Achievement
Test..................................................................................................15
Figure 6: Mother’s Achievement...................................................................................................16
Figure 7: Academic
Track..............................................................................................................17
Figure 8:
Ethnicity..........................................................................................................................18
Frequency Polygons.......................................................................................................................19
Figure 9: Math Achievement
Test..................................................................................................19
Figure 10: Mother’s Education......................................................................................................20
Figure 11:
Ethnicity........................................................................................................................21
Figure 12: Academic
Track............................................................................................................22
Descriptive Statistics......................................................................................................................23
Figure 13: Range, Standard Deviation, Skewness.........................................................................23
Frequency Table.............................................................................................................................24
Figure 14: Mean, Median, Mode...................................................................................................24
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT 4
SPSS Problems
Chapter 3
3.1. Bar chart: Why did/did not create one for each variable.
Utilizing rectangular bars of varying heights or lengths, a bar graph is an approach to data
visualization. This facilitates the presentation of pertinent statistics to the audience. The reader
can rapidly ascertain the quantity being contrasted through a cursory analysis of the bar's length.
The characteristics of the presented data determine whether a bar graph is horizontal or vertical.
To ensure the functionality of the data visualization, assign numerical values to the y-axis and
classify category variables along the x-axis. Barring a quantitative constant, bar graphs typically
accommodate only two variables. Visually representing a specific set of data is their intended
function. Shukla (2023) suggests that research could significantly benefit from employing a
multitude of bar graphs.
Bar Chart Discussion
Bar charts depicting data on academic performance and academic course exhibit
considerable variation in the two charts representing the mother's achievement in mathematics
and education. Conversely, the two charts denoting ethnicity and academic trajectory exhibit
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT 5
negligible variation. This analysis takes into account four distinct categories of variables: race or
ethnicity, the educational background of the mother, academic progression, and mathematical
aptitude. Academic discipline, as opposed to ethnicity and mathematics achievement, which are
both ordered variables, operates as a binary variable. When discussing names in relation to
classifications such as ethnicity or level of education, it is customary to employ the term
"nominal" (Morgan et al., 2020). This is due to the frequent usage of the expression to represent
names and its close association with them. Each of these categories has been assigned a distinct
set of numerical values in order to facilitate the assessment of the data collected within them.
Morgan et al. (2020) define the ordinal scale as a technique for assessing the pertinence of
individual features to the overarching subject matter. This is achieved by assigning a "rank" to
each feature in comparison to other characteristics.
1 See Figure 1
2 See Figure 2
3 See Figure 3
4 See Figure 4
3.2. Histograms: Why did/did not create one for each variable.
While bar charts and histograms share many similarities, the primary distinction is that
histograms exclude interbar spaces. The scores under consideration are derived from a
continuous variable along a continuum, which signifies that they may span the entire range from
the minimum to the maximum potential value. Given the absence of vacant spaces, it is probable
that a variable of this nature generates the scores. Continuous or non-continuous data may be
represented using histograms, provided that the underlying variable is also continuous. Although
the competency scale employed a four-point scale to assess its elements, Morgan et al. (2020)
highlight the theoretical possibility of possessing any level of expertise in any of the domains.
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT 6
Histogram Discussion
Among the supplied variables, a single one was deemed valid enough for the construction
of a histogram. This is due to the fact that histograms display the scores without any spaces
between the bars, which necessitates the use of a continuous variable to accurately depict the
data. The mathematics achievement exam is structured to incorporate interdependent
components, thereby allowing for a wide range of potential results on a scale spanning from
minimum to maximum. Histograms can be applied to non-continuous data when it is assumed
that the underlying variable is continuous. The following are the outcomes of the mathematics
achievement examination. Morgan et al. (2020) classify academic trajectory, ethnicity, and
mother's education as continuous variables as opposed to discrete factors. As a result, they argue
that employing a histogram to represent these components is inappropriate.
5 See Figure 5
3.3 (a). Frequency polygons: Why did/did not create one for each variable.
To facilitate the establishment of connections between nodes in data sets that contain a
wide variety of information, a frequency polygon is employed. Although the technique works
best with data that is approximately normal, it can also be applied effectively to ordinal data.
Visual depictions of continuous data are efficiently communicated via them, sharing
characteristics and operations with line graphs. Moreover, in order to facilitate comparisons
between two distinct distributions, they may be utilized (Morgan et al., 2020).
Frequency Polygons Discussion
It becomes evident that the graph representing the arithmetic achievement exam is the
only one that exhibits substantial content as we construct the graphs for this particular item (6).
In contrast to the others, the apex density of this one is greater. However, the academic trajectory
is a binary variable, while the ethnicity and level of education of the mother are nominal factors.
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT 7
Morgan et al. (2020) state that it is impracticable to construct frequency polygons using any of
the aforementioned factors.
3.4. Range, standard deviation, skewness: which are meaningful.
Range
Critically, variability is determined by comparing the disparity between the greatest and
lowest scores. Placing the scores in the order in which they were awarded provides a useful
indication of the extent of the range. A data error may be identified if the range of 1.3 in a
column contains the numeric values as well, which is not its intended function. This metric is
useful when searching for errors in data. As a result, this indicates that the initial dataset contains
a source of error (Morgan et al., 2020).
6 See Figure 9
7 See Figure 10
8 See Figure 11
9 See Figure 12
Standard Deviation
It involves the variance (x) of individual scores with respect to the
average of all scores. (∐x2) is subsequently utilized to compute the sum of
the squares of the deviation scores. The formula (SD = −∏x2/N-1) specifies
that the square root is acquired by dividing the sum by N-1 (Morgan et al.,
2020).
Skewness
The mean and median values of a frequency distribution differ despite the presence of a
larger number of stories. One can determine whether parametric or nonparametric methods
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT 8
should be implemented by substituting absolute skewness values for nonparametric statistics
(Morgan et al., 2020).
Range, Standard Deviation, Skewness Discussion
While the range remains unaffected by the educational attainment and ethnic background
of the mother, these factors significantly impact the academic progress and, to a lesser degree,
the proportion of students who achieve success in mathematics. Normal variables always
influence the standard deviation, but ordinal ranks have an even greater impact. When
performing the standard deviation calculation, nominal or dichotomous variables are ignored.
The skewness of a variable should not be heavily considered when dealing with nominal or
dichotomous variables. However, it is an imperative requirement when working with ordinal and
normal variables. According to Morgan et al. (2020), there is no correlation between skewness
and the results of the math achievement test in the case of the mother's ethnicity, educational
level, or academic accomplishments.
3.5. Mean, median, mode: which are meaningful.
Mean
Morgan et al. (2020) state that in the process of determining the central tendency of a
frequency distribution, the arithmetic mean incorporates all the easily accessible data.
Median
The median, also known as the middle value, is the most appropriate metric for representing
basic data at the ordinal level. Morgan et al. (2020) state that in situations involving an
asymmetrical distribution of frequencies, the median is regarded as a more dependable measure
of central tendency in comparison to the mean. This occurs due to the median, which signifies
the midpoint of a distribution.
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT 9
Mode
Although it typically provides the least precise information on central tendency, the most
prevalent category can be applied to any type of data (Morgan et al., 2020).
Mean, Median, Mode Discussion
In order to ascertain the central tendency of a distribution, three prevalent metrics can be
employed. This includes the mode, median, and mean as components. Data that adheres to a
normal distribution is amenable to analysis through any of the aforementioned techniques. It is
pointless to calculate the mean of the unprocessed values when conducting ordinal data analysis.
Conversely, an examination of the ranking scores as a whole reveals some intriguing insights
(Morgan et al., 2020). There is no statistically significant average when considering the
educational background or ethnicity of the mother. Nevertheless, it exhibits a certain degree of
academic relevance and exerts a statistically significant influence on the arithmetic achievement
test. The median does not exhibit a statistically significant correlation with the ethnicity or
educational attainment of the mother. Nevertheless, a correlation of statistical significance can be
observed between the median, academic performance, and course variables. Morgan et al.
(2020) establish that a correlation between the academic trajectory of her children and the
ethnicity and educational background of the mother is statistically significant. While the mother's
proficiency in mathematical operations can be altered, it is hardly a significant factor.
10 See Figure 14
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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References
Morgan, G. A., Barrett, K. C., Leech, N. L., & Gloeckner, G. W. (2020). IBM SPSS for
Introductory Statistics Use and Interpretation (6th ed.). New York, NY, USA: Routledge.
Shukla, D. (2023). A narrative review on types of data and scales of measurement: An initial step
in the statistical analysis of medical data. Cancer Research, Statistics, and Treatment
(Online), 6(2), 279-283. https://doi.org/10.4103/crst.crst_1_23
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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Figures - Figure 1: Mother’s Education
mother's
education
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a2ysvor
=
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nae
colored
omasters
MOD
co
mother's
education
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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Figure 2: Math AChievement Test
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te
math
achievement
test
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ES
math
achievement
test
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ep
ae,
Aan
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QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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Histograms - Figure 3: Ethnicity
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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Figure 4: Academic Track
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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Figure 5: Math Achievement Test
math
achievement
test
500
oo
5.00
10.00
1528
20.00
2.00
math
achievement
test
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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Figure 6: Mother’s Achievement
mether's
education
7”
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=
411
Sd
Gey,
=
224
Ha
TS
mother's
education
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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Figure 7: Academic Track
on
50
au
a
academic
track
“5
a
s
to
iS
academic
track
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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Figure 8: Ethnicity
Fi
ethnicity
re
dg?
aT
11
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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Frequency Polygons - Figure 9: Math Achievement Test
10
wnS
1m
oT
414.57
40
130
12.00
1oLB7
10.39
43
oo
math
achievement
test
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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Figure 10: Mother’s Education
Count
TLS.
14a.
grad
=
21S
vor
2yisvor
=
2yrs
> 2
Ts
call
coll
mother's
education
coll
grad
marsler's
MIDYPNS
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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Figure 11: Ethnicity
BUNA
mer
AlAtan-Amer
ethnicity
Labra.Armar
ASTER
ATH!
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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Figure 12: Academic Track
Count
a4
fast
track
academic
track
regular
track
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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Descriptive Statistics - Figure 13: Range, Standard Deviation, Skewness
Descriptive
Statistics
N Ranga
Mean
Std.
Deviation
Skeawness
Siatistic
Statistic
Slatisiic
Statistic
Siatisiic
Std.
Error
academic
track
5
1
55
501
«157
aif
mother's
education
75
8
4.14
2.240
1.424
27T
math
achievementtest
5
25,34
12.5645
6.67031
O44
iif
athnicity
73
3
1.77
1.021
1.052
281
Valid
NM
(listwise)
73
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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Figure 14: Mean, Median, Mode
|
ie
id
baat
math
mother’s
achievement
education
fest
academic
track
ethnicity
NW
Valid
75
rs iS ia
Missing
0 0
0
2
Mean
411
12.5645
55
177
Median
3.00
13.0000
1.00
1.00
Mode
3
14.33
1 1
QUANTITATIVE ANALYSIS: VARIABLES, Z SCORES, POPULATION, AND OUTPUT
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