Overview
Discussion Thread 2: Variables, Z Scores, Population, and Output
Jonathan Poppe
BUSI820: Quantitative Research Methods (B02)
Due 5/24/26
School of Business, Liberty University
Author Note
I have no known conflict of interest to disclose.
Correspondence concerning this article should be addressed to
Jonathan Poppe. Email: [email protected]
Discussion Thread 2: Variables, Z Scores, Population, and Output
If You Have Categorical, Ordered Data (Such as Low Income, Middle Income, High Income), What Type of Measurement Would You Have? Why?
Data that has a meaningful order or ranking without exact differences between categories would be an ordinal variable (Barrett et al., 2025). Ranking the data from low to high income would be meaningful, but without additional data, the gap between the various levels is unknown and likely uneven. It can’t be a nominal variable because the categories have a meaningful ranking, it can’t be a dichotomous variable because there are three categories instead of two, and it is not a scale variable because the categories have no measurable distance between levels of wealth.
D2.3.2
(A) Compare and Contrast Nominal, Dichotomous, Ordinal, and Normal Variables.
Nominal, dichotomous, ordinal, and scale (aka normal) variables are all methods used to classify and measure data, but they differ in how much information they provide and how the data can be analyzed (Barrett et al., 2025). A nominal variable uses categories with no natural order or ranking, like political party affiliation, religion, and eye color. These categories are simply labels, so a mathematical comparison between them gives no meaningful insight. A dichotomous variable is a specific type of nominal variable with only two possible categories, such as yes/no, pass/fail, or male/female.
An ordinal variable differs from a nominal variable in that the categories have a meaningful order or ranking (Barrett et al., 2025). For example, a set of satisfaction ratings like poor, fair, good, and excellent would be ordinal because the categories progress from lower to higher. However, because the exact distance between the categories is not equal or measurable, the satisfaction ratings are not a scale variable. Scale variables (aka normal variables) use numerical values with meaningful, measurable intervals between categories, such as age, weight, height, and temperature. Because the scale variables contain the most precise information among these four types, they allow for more advanced statistical analysis through measures such as means, standard deviations, and correlations.
(B) In Social Science Research, Why Isn’t it Important to Distinguish Between Interval and Ratio Variables?
In social science research, it is not necessary to distinguish between interval and ratio variables because the same statistical analysis can be applied to both without regard to which type of variable is being analyzed (Barrett et al., 2025). Technically, a ratio variable, such as temperature, has a true zero, whereas interval variables do not. Social science studies tend to concentrate on relationships, patterns, and differences in variables, rendering the existence of a true zero moot.
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?
Under a standard normal curve, 68.2% of the area is one standard deviation above or below the mean (Chatfield et al., 2026). This tells us that most scores in a normally distributed dataset will be fairly close to the average (Barrett et al., 2025). As a result, scores more than one standard deviation away from the mean will be relatively uncommon. Specifically, about 32% of the values will fall outside of that range, split across both tails of the distribution. A score more than one standard deviation away from the mean is unusual, and the further from the mean, the rarer that value will be.
D2.3.4.
(A) How do Z Scores Relate to the Normal Curve?
Z scores relate to the normal curve by expressing any raw score in terms of its position relative to the mean using standard deviations (Barrett et al., 2025). A z score tells you exactly how far a value is from the mean and in which direction. Positive z scores indicate values above the mean, negative z scores indicate values below the mean, and a z score of 0 is exactly at the mean.
(B) How Would You Interpret a Z Score of –3.0?
A z score of -3.0 means the value is three standard deviations below the mean (Chatfield et al., 2026). In a normal distribution dataset, this score is very unusual or rare, with only a small proportion of the scores being that low, probably less than .1% of the scores. Depending on the context and sample size, this observation is so far away from the other more typical values that it may be an outlier.
(C) What Percentage of Scores is Between a Z of –2 and a Z of +2? Why is this important?
About 95% of scores in a normal distribution fall between a z score of -2 and +2 (Chatfield et al., 2026).
This is important because it shows that most values in a normally distributed dataset cluster fairly close to the mean, within two standard deviations (Barrett et al., 2025). As a result, any score that falls outside this range (below -2 or +2) is relatively rare and can often be considered unusual or a potential outlier. This concept is used to identify typical vs atypical scores, assess probability, and make decisions about significance in hypothesis testing.
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?
You should not use frequency polygons with nominal data because they are designed for quantitative, ordered data with meaningful numerical values, such as age, IQ, or test scores (Barrett et al., 2025). The frequency polygon connects data points with lines to show trends on an ordered scale. However, nominal data include categories such as eye color, ice cream flavor preference, and religious affiliation, which have no inherent order. Connecting these categories with a line would imply a progression or relationship between them that does not actually exist.
A better method for displaying nominal data is a bar chart, where each category is represented separately and bar heights show frequencies or percentages without implying order or continuity between categories (Barrett et al., 2025). Another method to display nominal data is a pie chart, if the goal is to show proportions of categories within a whole. For example, the pie chart might show the percentage of the dataset that preferred vanilla ice cream vs. chocolate. These methods reflect that nominal data are distinct categories, not points on a scale.
ReferencesBarrett, K. C., Leech, N. L., Gloeckner, G. W., & Morgan, G. A. (2025). IBM SPSS for Introductory Statistics: Use and Interpretation (7th ed). Taylor & Francis Group.
Chatfield, M. D., Dobson, A. J., Marquart-Wilson, L., & Farewell, D. M. (2026). Characterizing the back-transformed standard deviation and closely related quantities. Communications in Statistics - Theory and Methods, 1–6. https://doi.org/10.1080/03610926.2026.2649788