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Nominal variables are used in statistics to label or categorize items without implying any numeric order or ranking among them. Each category or group is mutually exclusive and there's no inherent order to the categories. In the provided exercise, the examples of nominal variables are the type of tree used as a Christmas tree and the poll of registered voters as to which candidate they support. The trees are categorized by species, and candidates are categorized by name or political affiliation. These categories are qualitative and not quantitative, meaning they describe quality instead of quantity.

When analyzing nominal data, one may use mode as a measure of central tendency but cannot meaningfully compute a median or mean. Cross-tabulations and chi-square tests are typical methods used for analyzing such data.

Ordinal variables, unlike nominal variables, do have a clear order or ranking, but the intervals between the values are not necessarily equal or known. These variables are often found in survey research where responses can rank preferences or levels of agreement. For instance, educational levels (e.g., elementary, middle school, high school, college) represent an ordinal variable; there is a clear order from lower to higher education, but the difference between each level is not the same nor based on a consistent scale.

Ordinal data can be analyzed using non-parametric tests which do not assume uniform intervals, such as Spearman's rank correlation or the Kruskal-Wallis test. These tests help determine relationships and differences when dealing with ranked data.

Discrete variables are countable in a finite amount of time and can only take on certain values. In statistics, they are the backbone for situations where we count occurrences, such as the number of televisions within a household or the number of pages printed by a computer printer, as in the exercise examples. These values are integers and gaps exist between the numbers; you can't have half a television or print three-quarters of a page.

In analyses, discrete variables may lead us to use probability mass functions and can be described using frequency distributions. Measures such as mode and median are also applicable. For more detailed examination, one might employ Poisson or binomial distributions, particularly when modeling the likelihood of a certain number of events occurring within a fixed time or space.

Continuous variables can assume any value within a given range and are not limited to distinct whole numbers. They are often measurements, like lengths or times. The exercise's examples include the time required for a wound to heal with new medicine or the distance a football is kicked, both of which can vary to a fine degree and include fractions or decimals. Because the possible values are infinite, continuous variables provide a rich source of nuanced data.

For such variables, statisticians use probability density functions, and measures of central tendency include the mean, median, and sometimes the mode. In-depth statistical analysis might involve using normal distributions or applying tests like ANOVA (analysis of variance) to discern differences between groups or test for certain trends over a continuous range.