Discussion Forum 2: Variables, Z Scores, Population and Output
Lucianna Easton
BUSI 820: Quantitative Research Methods
January 26, 2025
Author Note
I have no known conflict of interest to disclose.
Correspondence concerning this article should be addressed to Lucianna Easton. Email:
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?
Based on the research by Morgan et al., 2020, the measurement type in question can be
classified as3ordinal. The key factor to consider here is the ordered nature of the data. It's
important to note that the data does not fit into the categories of interval or nominal. This is due
to the likely unequal distances between the categories of low, middle, and high income.
Furthermore, the notion of being approximately nominal is typically associated with five or more
ordered categories. The term 'categorical' is often linked with 'nominal', which can make this a
bit complex. However, if the categories have an inherent order, the variable should be identified
as3ordinal.
D2.3.2 (a) Compare and contrast nominal, dichotomous, ordinal, and normal variables.
Nominal variables3represent the most basic level of measurement. In this case, the
numerals assigned to each category merely serve as identifiers, without any inherent order or
value (Morgan et al., 2020). While the use of only nominal variables can limit the statistical
methods applicable to the data, it does not completely eliminate the potential for statistical
analysis. Therefore, even if the data falls into unordered or nominal categories, appropriate
statistical methods can still enhance the research. Dichotomous variables, as explained by
Morgan et al. (2020), are those that have only two levels or categories at any given time. Some
of these variables may have an implied order, while others do not. There are several benefits to
using precise statistics for dichotomous and nominal variables. Moving up the scale,3ordinal
variables3are not only mutually exclusive, like nominal variables, but they also have an order,
ranging from low to high, which allows for ranking. The numbers on the ordinal scale indicate a
higher rank or quantity, but the intervals between different categories are not equal (Morgan et
al., 2020). Ordinal variables consider whether the frequency counts for each category or value
are distributed in a bell-shaped, normal distribution, with more responses in the middle
categories and fewer in the lowest and highest categories.
D2.3.2 (b) In social science research, why isn’t it important to distinguish between interval
and ratio variables?
The main objective in social science research is to ascertain the consistent distribution of
the independent variable (Morgan et al., 2020). Interval and ratio scales are differentiated by
their ability to register values below zero. Interval scales, lacking a true zero, can represent
negative values and measure all quantitative attributes. They allow for ranking, counting,
subtraction, and addition, with equal intervals between each number. However, this does not
indicate any ratio relationship between the numbers (Morgan et al., 2020). Ratio scales share
these properties but are distinguished by the presence of a true zero-point, known as the property
of origin.
D2.3.3 What percent of the area under the standard normal curve is within one standard
deviation of (above or below) the mean? What does this tell you about scores that are more
than one standard deviation away from the mean?
The exact percentage is334.13%, which could be roughly 34% higher or lower. About
32% of the scores deviate more than one standard deviation from the mean in a normally
distributed data set (Morgan et al., 2020). In a normally distributed data set, approximately 68%
of the data values fall within one standard deviation of the mean. This includes the regions to the
right and left of the mean (Morgan et al., 2020). Two standard deviations to the right of the mean
cover about 47.5% of the area under the normal curve. Two standard deviations to both sides of
the mean cover approximately 95% of the area under the standard curve (Morgan et al., 2020).
D2.3.4 (a) How do z scores relate to the normal curve?
The standard normal distribution unit, also known as the z-score, is found under the
normal distribution. The standard normal curve is a transformation of the normal curve, achieved
by setting the mean to zero and the standard deviation to one. In any statistics book, the normal
curve table provides the areas under the curve for one standard deviation (z = 1) and two
standard deviations (z = 2) (Morgan et al., 2020). This transformation allows for the comparison
of normal curves with different means and standard deviations (Morgan et al., 2020). All normal
curves share the same proportions within one, two, or three standard deviations.
D2.3.4 (b) How would you interpret a z score of –3.0?
The z-score is positive if
the value is more than
the mean and negative if
it is less than the
mean. The z-score value
indicates how many
standard deviations you
are from the mean (Song
et
al., 2019). If the z score
is -3.0, it means that it is
three standard
deviations below the
mean
(Morgan et al., 2020)
As outlined in this week assigned readings, per Morgan et al., 2020, the z-score is
positive if the value is more than the mean and negative if it is less than the mean. The z-score
value indicates how many standard deviations you are from the mean. If the z score is -3.0, it
means that it is three standard deviations below the mean.
D2.3.4 (c) What percentage of scores is between a z of –2 and a z of +2? Why is this
important?
Researchers can use a primary normal distribution to calculate the likelihood of randomly
obtaining a score from the distribution or sample. For example, there is a 68 percent chance of
selecting a score between -1 and +1 standard deviations from the mean at random. In a normal
distribution, approximately 95% of the scores fall between a z of -2 and a z of +2. This is
significant since the statistical findings for the population are derived from these scores (Morgan
distribution,
approximately 95% of
the scores fall between a
z of -2 and a z of +2. This
is
significant since the
statistical findings for
the population are
derived from these
scores (Morgan
et al., 2020).
D2.3.5 Why should you not use a frequency polygon if you have nominal data? What would
be better to use to display nominal data?
A frequency polygon is a graphical representation of the distribution of a dataset. It is an
improvement over a histogram which helps to visualize the shape of the distribution. However, it
is not suitable for nominal data. Nominal data is a type of data that is used to name variables
without providing any numerical value. It is the simplest form of a scale of measure and is not
suitable for a frequency polygon because nominal data is categorical and does not have a logical
order (Morgan et al., 2020). Frequency polygons require a logical sequence or numerical data. In
addition, nominal data cannot be used for calculations. Frequency polygons often involve
calculations such as finding the mean or median. Bar graphs and pie charts are more suitable
alternatives to displaying nominal data and frequency associated with an applicable category.
References
Morgan, G. A., Barrett, K. C., Leech, N. L., & Gloeckner, G. W. (2020).3IBM SPSS for
introductory statistics: Use and interpretation3(6th ed.). Routledge.
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