DISCUSSION THREAD: VARIABLES, Z SCORES, POPULATION AND OUTPUT 1
Marcos Reis Campos, MBA
BUSI 820
Liberty University
May 26, 2024
Table of Contents
DISCUSSION THREAD: VARIABLES, Z SCORES, POPULATION AND OUTPUT 2
Variables, Z Scores, Population and Output ...................................................................................3
Legend .............................................................................................................................................3
Discussion Board Chapter 3 Questions ...........................................................................................3
D2.3.1. If you have categorical, ordered data (such as low income, middle income, high income)
what type of measurement would you have? Why? .......................................................................4
D2.3.2. (a) Compare and contrast nominal, dichotomous, ordinal, and normal variables..............4
Nominal
Variables............................................................................................................................4
Dichotomous Variables ...................................................................................................................4
Ordinal Variables ............................................................................................................................4
Normal Variables ............................................................................................................................6
Compare/Contrast ...........................................................................................................................6
D2.3.2. (b) Do interval and ratio variables matter in social science research?................................7
D2.3.3. (a) What % of area under standard normal curve is within one σ µ?.................................7
D2.3.3. (b) Significance for scores a σ or more from µ?.................................................................7
D2.3.4. (a) How do z scores relate to the normal curve?.................................................................7
D2.3.4. (b) How would you interpret a z score of –3.0?..................................................................8
D2.3.4. (c) What percentage of scores is between a z of –2 and a z of +2?....................................8
D2.3.4. (c) Why is this important?...................................................................................................8
D2.3.5. (a) Why not use a frequency polygon with nominal data?.................................................8
D2.3.5. (b) What is the best way to display nominal data?.............................................................8
References......................................................................................................................................10
DISCUSSION THREAD: VARIABLES, Z SCORES, POPULATION AND OUTPUT 3
Variables, Z Scores, Population and Output
Legend
µ = mean
σ = standard deviation
Discussion Board Chapter 3 Questions
D2.3.1. If you have categorical, ordered data (such as low income, middle income, high
income) what type of measurement would you have? Why?
Because categorized data is essential for resolving the issue, ordinal measurements are
suitable for implementation in this circumstance. Ordinal variables are used to represent
alternatives when individuals must select them from a system of clearly defined categories. Data
denoted by the term "ordinal data" consists of values arranged in a distinct and understandable
hierarchy. Elevated distances between the low-, middle-, and high-income groups account for the
lack of interval or normal characteristics in the data. This is the case, notwithstanding the
undeniable fact that low income is intermediate to medium income. Moreover, at least five
categorized categories must be present for the data to show a satisfactory level of normality.
When discerning between nominal and categorical terms, it is prudent to treat the categories in a
particular sequence as ordinal variables, as this clarity can be restored.
Ordinal and categorical variables share a similar characteristic in that they both consist of
sequential categories. Conduct an examination of the economically significant variables
incorporated. Low, medium, and high-income levels should be used to classify them into three
distinct categories. While it is feasible to arrange the numbers in ascending order, the factor that
influences the degree of variation in the distance between them is the variable level. Moving
forward, we shall analyze the disparities in income levels present across the distinct categories.
DISCUSSION THREAD: VARIABLES, Z SCORES, POPULATION AND OUTPUT 4
There will be an increased magnitude of discrepancy between the two categories. The variable
under consideration would be classified as an interval if the categories were distributed in an
even fashion (Baak et al., 2020).
D2.3.2. (a) Compare and contrast nominal, dichotomous, ordinal, and normal variables.
Nominal Variables
A categorical level of the variable, consisting of qualitative categories that are
incompatible with one another, was identified in a study by Morgan et al. published in 2020. A
nominal variable is characterized by the absence of an inherent hierarchy or order among its
constituent categories. The term "binary variable" refers to a category variable consisting of two
distinct categories. A yes/no inquiry serves as an illustration of a binary variable. No intrinsic
factors determine the ranking or sequence of these categories. Determining the relative
importance of the various classifications of hair color is a subject that defies agreement. It is
feasible to subject variables that are entirely nominal, meaning the categories cannot be precisely
organized within the variable, to categorization (Baak et al., 2020).
Dichotomous Variables
According to Morgan et al. (2020), a two-level variable is frequently represented in SPSS
as zero and one. Consisting of only two levels, these variables are the most fundamental.
Ordinal Variables
In addition to possessing an ordered nature, obtain a variable that frequently displays
skewness (Morgan et al., 2020). Among the many categorical variables, ordinal variables are
characterized by their sequential arrangement. The economic status variable consists of three
discrete classifications, as explained in the initial inquiry: low income, medium income, and high
income. Consider the subsequent variable. Following this, we shall apply this notion to a variable
DISCUSSION THREAD: VARIABLES, Z SCORES, POPULATION AND OUTPUT 5
representing educational achievement, which comprises the subsequent values: bachelor's
degree, high school completion, elementary school graduation, some college completion, and
college completion. The degree of the variable may induce a variation in the distance between
the numbers, despite the possibility that they could be arranged in a higher order. A thought-
provoking endeavor entails conducting a comparative analysis of the educational backgrounds of
individuals classified as belonging to categories one and two, two and three, or three and four. A
stronger correlation is probable between categories one and two (primary and secondary
education) in comparison to categories two and three (high school and a fraction of higher
education). Grouping individuals according to their educational attainment is the situation
presented. As the distance between categories one and two is significantly greater than the gap
between categories two and three, the disparities between the groups are, nevertheless,
inconsequential. The variable being examined would be classified as an interval if the categories
were distributed in an equitable manner (Baak et al., 2020).
Normal Variables
Frequently encountered terminology includes "interval/ratio variable" in textbooks and
the statistical software SPSS (Morgan et al., 2020). The standard normal random variable of the
standard normal distribution. A random variable that adheres to a normal distribution and
possesses a mean (µ) of zero and a standard deviation (σ) of one is denoted as such. To denote
the precise location, the letter Z must be consistently employed (Eidous & Abu-Shareefa, 2020).
Compare/Contrast
Real-valued random variables are regulated by a normal distribution, which is classified
as a continuous probability distribution, within the realm of probability theory. The parameters ε
and σ, which represent the expectation value and standard deviation of the distribution,
DISCUSSION THREAD: VARIABLES, Z SCORES, POPULATION AND OUTPUT 6
respectively, establish the probability density function. Exclusive to ratio scales is one that
features a true zero point. Separating adjacent values with equal intervals is possible when
interval and ratio data are applied sequentially. Due to the lack of a minimum temperature of
zero, the Fahrenheit (or Celsius) scale is an interval scale. A value of zero is assigned to the
absence of heat on the Kelvin scale. Variables are mutable entities that are susceptible to
perturbation, control, and measurement. Qualitative or discrete variables are sometimes
employed in place of categorical variables (Morgan et al., 2020). In the category of variables,
nominal, ordinal, and dichotomous variables are included.
When no inherent order is present, nominal variables are defined as sets of two or more
categories. Apartments, bungalows, cooperatives, and houses are just a few of the many
designations that real estate agents can apply to their clients' properties. The nominal variable
"property type" comprises four discrete categories: homes, apartments, cottages, and
cooperatives. The designation "diamond variable" is applied to a variable that possesses an
absolute minimum of two values. To provide an example, within the realm of gender, people are
conventionally categorized as either "male" or "female." Investigating whether an individual is in
possession of a cellular device could serve as an additional example. The ownership of a mobile
phone can be categorized as either "yes" or "no." Ordinal variables, which are like nominal
variables, consist of two or more sortable categories. An ordinal variable is apparent in the
response options provided by individuals when examining their position on the policies of the
Democratic Party: "not at all," "excellent," or "very much." Based on your establishment of three
discrete categories—namely "not many," "they are fine," and "the smallest"—and the fact that
"values" cannot be allocated to ranked levels, this approach is deemed impracticable (Morgan et
al., 2020).
DISCUSSION THREAD: VARIABLES, Z SCORES, POPULATION AND OUTPUT 7
D2.3.2. (b) Do interval and ratio variables matter in social science research?
Social science researchers must possess an exhaustive comprehension of the distinctions
between interval and ratio data. In contrast to interval data, ratio data possesses a solitary zero
point. Regrettably, interval data fails to satisfy this stipulation. It is essential that both interval
and ratio data be covered, as both are utilized in social science research. Verifying the efficacy of
empirical social science research requires the application of precise and reliable measurements.
Social science researchers assess intangibles that are difficult to quantify, such as beliefs,
perspectives, attitudes, and perceptions, as part of their data collection methodology. Scholars
specializing in the social sciences frequently commit the error of data collection lacking
sufficient specificity. According to Morgan et al. (2020), while it is always possible to convert
data from one higher-level structure to another, the converse is not always possible.
D2.3.3. (a) What % of area under standard normal curve is within one σ µ?
Morgan et al. (2020) estimate that the distance between µ and one σ on a normal curve is
approximately 34%.
D2.3.3. (b) Significance for scores a σ or more from µ?
A quarter-final result, comprising approximately 68% of the data, will be within one-half
of the µ, 95% will be within three σs of the µ, and 99.7% will be within three-half of the µ , in
that order. According to Morgan et al. (2020), every measurement that follows the normal
distribution may be measured using ξ, which is also known as the standard error of the µ.
D2.3.4. (a) How do z scores relate to the normal curve?
The z-score is a metric that indicates the proximity of a raw score to the mean (µ) through
the calculation of its distance in standard deviation units (σ) from the mean. According to
DISCUSSION THREAD: VARIABLES, Z SCORES, POPULATION AND OUTPUT 8
Morgan et al. (2020), this notion is exclusive to the normal distribution and is not universally
applicable to other distributions.
D2.3.4. (b) How would you interpret a z score of –3.0?
Three σs below the µ
D2.3.4. (c) What percentage of scores is between a z of –2 and a z of +2?
An estimated ninety-five percent of the data occurs between two σs of the µ.
Consequently, 95% of the data ranges from -2 to +2 on the z-score scale.
D2.3.4. (c) Why is this important?
The application of a standard normal distribution enables scientists to ascertain the
likelihood of arbitrarily obtaining a score from a given distribution or sample. To illustrate,
selecting a score at random that is within one standard deviation (σ) of the mean (µ) in the given
distribution has a 68% chance of occurring. 95% of the data points fall within the z-score range
of -2 to +2, which corresponds to the interval of scores within two standard deviations of the
mean. Assuming a normal distribution, this range is indicative thereof. This is significant
because statistical inferences regarding the population are derived from these scores (Morgan et
al., 2020).
D2.3.5. (a) Why not use a frequency polygon with nominal data?
Because they assume a seamless transition between data points, frequency polygons may
not be suitable for analyzing ordinal or nominal data (Morgan et al., 2020).
D2.3.5. (b) What is the best way to display nominal data?
Hypotheses based on nominal data can be assessed through the utilization of
nonparametric tests, such as the chi-square test (Morgan et al., 2020). The objective is to
DISCUSSION THREAD: VARIABLES, Z SCORES, POPULATION AND OUTPUT 9
determine whether the difference between the predicted and actual frequencies of values in a
dataset is substantial enough to be considered statistically significant.
References
DISCUSSION THREAD: VARIABLES, Z SCORES, POPULATION AND OUTPUT 10
Baak, M., Koopman, R., Snoek, H., & Klous, S. (2020). A new correlation coefficient between
categorical, ordinal and interval variables with pearson characteristics. Computational
Statistics & Data Analysis, 152, 107043. https://doi.org/10.1016/j.csda.2020.107043
Eidous, O. M., & Abu-Shareefa, R. (2020). New approximations for standard normal distribution
function. Communications in Statistics. Theory and Methods, 49(6), 1357-1374.
https://doi.org/10.1080/03610926.2018.1563166
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.
