2.1 Data Sets and What Data “Looks” Like

By now you have learned that variables are tools that allow us to measure concepts and to collect information about them. As such they are comprised of information — information that varies across the units of analysis (the ‘things’ on which we collect information, be it people, organizations, countries, etc.). So far, we have discussed individual variables – but creating and collecting information on a single variable is uncommon. Generally, we collect information on many variables at the same time (which, in turn, allows us to analyze variables together and hypothesize about possible associations between variables).

Variables “live” in data sets (or datasets, as I prefer; both usages are common). A dataset is a collection of variables that lists the information (or observations) gathered on them from the units of analysis. As usual, I focus on analysis of people for simplicity’s sake (but do keep in mind the units of analysis can be something else.)

The best way to visualize a dataset is as a sort of a table (a.k.a a matrix) which summarizes the responses from every individual (in the rows of the table) on the variables in the dataset (in the columns of the table). As such, the size of a dataset depends on two things: the number of variables and the number of individuals supplying information (a.k.a. respondents). Typically, datasets vary in size from just a handful of variables and few respondents to hundreds of variables and thousands of respondents. (Huge datasets — comprising information on millions of people — exist too; these are known as big data. Big data is not analyzed in the conventional ways regular datasets are, so from now on we’ll leave big data aside as it’s not the subject of this book.)

To start small, imagine you have just four friends at your university and you decide to list some items of information about them (say, maybe you want to compare your standing at the university with theirs, and to see differences and commonalities between you and them). You could do that in a sentence form, for example, thus: Arjun, who is twenty years old, speaks Punjabi at home and is a first year student in the Business School, has a job and his GPA is 3.6. Benjamin, on the other hand, who is 25, speaks German at home and is a third year Science student, also has a job but his GPA is lower than Arjun’s at 3.2. Cecilia, who speaks Spanish at home and is a fourth year Health Sciences student doesn’t have a paying job and her GPA is the highest of your friends, 4.0. Finally, Xingxing is also a first year student and is employed like Arjun but she is an Arts major, speaks Mandarin at home, and her GPA is 3.3.

Indeed, you might do that but the points of comparison might get lost as they are not easy to see: one has to read very carefully to keep track of who does what and has a GPA of how much. Instead, you could present the same information as it is in the table in Example 2.1 below.

If you do that, what you have created is a dataset. Now imagine that instead of this contrived combination of four friends and their varying characteristics, I generalize the example like so:

In Example 2.1 (B), the respondents are the four people on whose varying characteristics we have information, and these are represented by the six variables. This, however, seems rather cumbersome. Instead of “Variable 3”, and “Respondent 5”, and “Response_4.3“, etc., a simpler way to represent all of these in a generalized way is through mathematical notation.^[1]

So, prepare yourselves! Here comes notation:

In Example 2.1 (C), I₁, I₂, I₃, and I₄ are the four individuals; X₁, X₂, X₃, X₄, X₅, and X₆ are the six variables; and x₁₁, x₁₂, etc. stand for any specific characteristic/response a respondent has on a variable. More specifically, x₅₃, for example, is the characteristic that Respondent #3 has on Variable 5. Scrolling up to Example 2.1 (A) will allow you to see that x₅₃ is Health, which is Cecilia’s Major by Faculty.

From here, it’s not difficult to extrapolate the specific dataset we had above to a general one. Thus, Example 2.1 (D) below presents a template of a typical dataset.

In the table above, you may think of N as the last row on the table, i.e., the last individual for whom we have information and you may think of K as the last column on the table, i.e., the last variable we have in the dataset. Both numbers can theoretically be “any positive number”, though in practice the former is usually a number up to several thousands and the latter a number up to few hundreds. The ellipses in the next-to-last row and the next-to-last column indicate that the table is truncated:   there are omitted rows between the seventh and the last individuals (i.e., between I₇ and I_N), and omitted columns between the seventh and the last variables (i.e., between X₇ and X_K). (They obviously have to be omitted so that the table can fit on the page.)

Armed with this knowledge, let’s take a look at an excerpt from a real dataset. The following Example 2.1 (E) provides a snapshot of the first ten respondents and first nine variables in the Aboriginal Peoples Survey 2012 dataset (or APS 2012 for short)^[2] using a software called IBM® Statistical Package for the Social Sciences, commonly referred to as SPSS. 

One thing you might find surprising is the obvious fact that all cell entries (i.e., the observations we have) are listed in a number format. Does that mean that all variables in this particular dataset are interval or ratio? What about any nominal or ordinal variables – do they not exist in this dataset? The answer is no on both accounts: the variable SEX (i.e., “Sex of respondent” as stated in Variable View) is nominal and the variable AGE_YRSG (i.e., “Age group of respondent…”) is ordinal because of the hierarchical arrangement of the responses.  However, the dataset cells contain only numbers because statistical software can only analyze numerical data.

To that effect, nominal and ordinal variables appear “in code” in datasets; i.e., the categories of nominal and ordinal variables are assigned numerical values as labels to represent them in the actual dataset you might be working with. Thus, the numbers in nominal and ordinal variables’ columns are not actual numbers, they are artificially (and in the case of nominal variables, somewhat arbitrarily) assigned to represent the words contained in the categories in order to make computer-based statistical analysis possible. (On the other hand, interval/ratio variables’ categories contain actual numbers. Of course, the trick then is to learn to differentiate the actual numbers from the code/ number values used as labels in the cells of a dataset.)

Therefore, you should always keep track of the code (see the Watch Out! panel below for tips on Variable View in SPSS which allows you to do that), and remember to refer to the categories by their proper (word-based) names — not by the artificial numerical values (i.e., code) representing them!