VARIABLES, Z SCORES, POPULATION AND OUTPUT
Quantitative research relies heavily on numerical data and statistical methods to draw
meaningful conclusions (Creswell, 2014). Central to this approach are variables, which represent
different attributes or qualities that can be measured or observed (Babbie, 2009). Variables can
be manipulated or simply observed to understand relationships and patterns. Another essential
concept is the z-score, a statistical measurement that describes a value's relationship to the mean
of a group of values (Gravetter & Wallnau, 2016). It is instrumental in understanding how
individual data points relate to a larger distribution. Population refers to the group a researcher
wishes to study or draw conclusions about (Frankfort-Nachmias & Leon-Guerrero, 2017).
Quantitative research results, often presented in tables, graphs, or other visual representations,
constitute the output, providing a clear and concise way to interpret and understand the data. This
discussion will delve deeper into these foundational concepts, exploring their significance and
application in quantitative research.
Short Answer 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?
- - -The measurement is ordinal. This is primarily because the data exhibits a clear order or
hierarchy. For instance, low income is undeniably less than middle income. However, it's crucial
to note that this data doesn't fit the interval or normal measurement criteria. This is because the
gaps between low, middle, and high-income categories might not be consistent or equal.
Furthermore, for a data set to be approximately normal, it should encompass five or more
ordered categories (Morgan et al., 2020; Stevens, 1946). The terminology can be perplexing,
especially since "categorical" is frequently linked with "nominal" data. Yet, when there's an
inherent order in the categories (Morgan et al., 2020), it's more appropriate to categorize the
variable as ordinal (Agresti, 2010; Morgan et al., 2020).
D2.3.2.
(a) Compare and contrast nominal, dichotomous, ordinal, and normal variables.
Nominal variables are categorical and have two or more categories without inherent order or
hierarchy. For instance, variables like gender (male, female) or hair color (black, brown, blonde)
are nominal. The critical characteristic of nominal variables is that they are purely categorical,
with no inherent order among the categories (Velleman & Wilkinson, 1993). A specific subtype
of nominal variables, dichotomous variables, have only two distinct categories or outcomes.
Examples include outcomes such as Pass/Fail or Yes/No. While they are a subset of nominal
variables, the distinguishing feature is their binary nature (MacCallum et al., 2002). Ordinal
variables stand out because they represent categories with a specific order or rank. However,
while there's a clear hierarchy, the distances between these ranks or orders are inconsistent or
uniform. For instance, socioeconomic status (with categories like low, middle, and high) or
education level (ranging from high school to Ph.D.) are ordinal. The categories have a clear
order, but the intervals between them aren't consistent (Agresti, 2010). Normal variables, referred
to by Morgan et al. (2020) as approximately normal or scale variables, are roughly normally
distributed and not only exhibit an order in their levels or scores from the lowest to the highest,
but their score frequencies also tend to follow a near-normal distribution.
- - -In contrasting these variable types, the key differences lie in order, intervals, and categories.
While nominal and dichotomous variables lack an inherent order, ordinal and normal variables
possess it. Among them, only normal variables have consistent intervals. Dichotomous variables
are unique in having just two categories, whereas the others can have multiple. Lastly, only ratio
variables, a subset of normal variables, boast a true zero point.
(b) In social science research, why isn’t it important to distinguish between interval and ratio
variables?
- - -Most statistical procedures in SPSS treat interval and ratio data types similarly, not
differentiating between the two scales during computations (Field, 2013). The responsibility of
interpreting results, including discerning the nature of the data, primarily lies with the researcher
rather than the software. This means that understanding whether a variable possesses a true zero
(as in ratio scales) or lacks it (as in interval scales) is more about theoretical comprehension than
computational distinction (Pallant, 2016). Furthermore, when inputting and managing data, SPSS
doesn't require users to explicitly label a continuous variable as interval or ratio. The focus is on
differentiating between categorical (nominal and ordinal) and continuous data (George &
Mallery, 2019).
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?
- - -Approximately 68%, 34% above and 34% below, of the area under the standard normal curve
is within one standard deviation above or below the mean. This tells us that about 68% of the
data points in a normally distributed dataset fall within one standard deviation of the mean.
Consequently, scores that are more than one standard deviation away from the mean are in the
remaining 32% of the distribution. This 32% is split between the two tails, so about 16% of
scores lie more than one standard deviation above the mean, and another 16% lie more than one
standard deviation below the mean. In practical terms, if a score is more than one standard
deviation from the mean in a normally distributed dataset, it is relatively less common or less
frequent than scores closer to the mean.
D2.3.4.
(a) How do z scores relate to the normal curve?
- - -A z-score provides information about how many standard deviations a particular data point
(or score) is from the mean of a distribution. A z-score of 0 indicates that the data point is at the
mean. A z-score of 1 means the data point is one standard deviation above the mean. A z-score
of -1 indicates that the data point is one standard deviation below the mean.
(b) How would you interpret a z score of –3.0?
- - -A z-score of -3.0 indicates that the data point (or score) is three standard deviations below the
mean. In a standard normal distribution, where the mean is 0, this score would be positioned at a
value of -3 on the horizontal axis.
(c) What percentage of scores is between a z of –2 and a z of +2? Why is this important?
- - -Approximately 95% of the scores lie between a z-score of -2 and a z-score of +2 in a standard
normal distribution. According to Morgan et al. (2020), when you deduct this 95% from 100%,
the residual 5% corresponds to the commonly used probability or p-value of 0.05, a threshold for
determining statistical significance. Scores that don't fall within this two standard deviation range
are considered uncommon occurrences.
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?
- - -Frequency polygons are best suited for data that is normally distributed or on a scale, as they
represent data values as progressing continuously across the chart. Frequent distributions or bar
charts are more appropriate visual representations for nominal data. While frequency polygons
References
Agresti, A. (2010).-Analysis of Ordinal Categorical Data.-Wiley.
Babbie, E. (2009).-The Practice of Social Research.-Wadsworth Publishing.
Creswell, J. (2014).-Research Design: Qualitative, Quantitative and Mixed Methods
Approaches.-Sage Publications.
Field, A. (2013).-Discovering Statistics Using IBM SPSS Statistics.-Sage Publications.
Frankfort-Nachmias, C., & Leon-Guerrero, A. (2017).-Social Statistics for a Diverse
Society.-Sage Publications.
George, D., & Mallery, P. (2019).-IBM SPSS Statistics 26 Step by Step: A Simple Guide and
Reference.-Routledge.
Gravetter, F., & Wallnau, L. (2016).-Statistics for the Behavioral Sciences.-Cengage Learning.
MacCallum, R., Zhang, S., Preacher, K., & Rucker, D. (2002). On the practice of
dichotomization of quantitative variables.-Psychological Methods, 19-40.
Morgan, G., Barrett, K., Leech, N., & Gloeckner, G. (2020).-IBM SPSS for Introductory
Statistics: Use and Interpretation Sixth Edition.-Routledge.
Pallant, J. (2016).-SPSS Survival Manual.-Open University Press.
Stevens, S. (1946). On the theory of scales of measurement.-Science,
https://doi.org/10.1126/science.103.2684.6.
Velleman, P., & Wilkinson, L. (1993). Nominal, ordinal, interval, and ratio typologies are
misleading.-The American Statistician, 65-72.
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