Part I: Ways to Describe Data

In the introduction to this case study we identified four contrasting ways to describe data: categorical vs. numerical, ordered vs. unordered, absolute reference vs. arbitrary reference, and discrete vs. continuous. To give meaning to these descriptive terms, let’s consider the data in Table 1.

Table 1. Distribution of Yellow and Red M&Ms

bag id	year	weight (oz)	type	# yellow M&Ms	% red M&Ms	total M&Ms	rank (total M&Ms)
a	2006	1.74	peanut	2	27.8	18	sixth
b	2006	1.74	peanut	3	4.3	23	fourth
c	2000	0.80	plain	1	22.7	22	fifth
d	2000	0.80	plain	5	20.8	24	third
e	1994	10.0	plain	56	23.0	331	second
f	1994	10.0	plain	63	21.9	333	first

The entries in Table 1 are organized by column and by row, with the first row (shaded here for emphasis) identifying the variables used to describe the data. Each additional row is the record for one sample and each entry in a sample’s record provides information about one of its variables; thus, the data in the table lists the result for each variable and for each sample.

Investigation 1.

Of the variables included in Table 1, some are categorical and some are numerical. Define these terms and assign each of the variables in Table 1 to one of these terms.

Investigation 2.

Suppose we decide to code the type of M&M using 1 for plain and 2 for peanut. Does this change your answer to Investigation 1? Why or why not?

Investigation 3.

Categorical variables are described as nominal or ordinal. Define the terms nominal and ordinal and assign each of the categorical variables in Table 1 to one of these terms.

We also can use a numerical variable to assign samples to groups. For example, we can divide the plain M&Ms in Table 1 into two groups based on the sample’s weight. What makes a numerical variable more interesting, however, is that we can use it to make quantitative comparisons between samples; thus, we can report that there are 14.8× as many plain M&Ms in a 10-oz. bag as there are in a 0.8-oz. bag. Although we can complete meaningful calculations using any numerical variable, the type of calculation we can perform depends on whether or not the variable’s values have an absolute reference.

Investigation 4.

A numerical variable is described as either ratio or interval depending on whether it has (ratio) or does not have (interval) an absolute reference. Explain what it means for a variable to have an absolute reference and assign each of the numerical variables in Table 1 as either a ratio variable or an interval variable. Why might this difference be important?

Finally, the granularity of a numerical variable provides one more way to describe our data.

Investigation 5.

Numerical variables also are described as discrete or continuous. Define the terms discrete and continuous and assign each of the numerical variables in Table 1 to one of these terms.