ResearchMethods/QuantMix/Measurements

= Introduction =

When data are prepared for analysis by computer, values of variables are usually entered as numbers. Sometimes thsi form of data makes sense — for example, the population of a country or the number of votes received by a candidate. Sometimes, artificial numerical codes need to be created for convenience of processing. In a file containing data on members of Federal Australian parliament, for example, Liberals might be coded numerically as 1, Labour as 2, and Greens as 3. Numerical, however, is not the same thing as quantitative. In fact, whether data are coded numerically or not, there are different levels of measurement.

= Nominal Data =

If the variable is nominal, it means that the values do not indicate the amount of the thing being measured, or that the variables are in any particular order. Most of the time, there is no meaning to the numbers: they are chosen at random, in an arbitrary way. For example, if we list the states and territories of Australia, we are not indicating the amount of "stateness" each possesses, nor listing them in order of "stateness." We may code the states as "1," "2," "3," and "4" respectively, but this is just for convenience, and in no way quantifies what we are doing. Each value, numerical or otherwise, is merely a label or name (hence the term "nominal").

= Ordinal Data =

Sometimes the values of a variable are listed in order. Or we could say that the values are "ranked" or "rank ordered." For example, the army orders (ranks) military personnel from general to private. At a university, class standing of undergraduates (first to fourth year) is another example of an ordinal variable. In both of these examples, the values of the variable in question (military rank or class standing) are ranked from highest to lowest or vice versa. There are other kinds of ordering. For example, respondents in a survey may be asked to identify their political activism as "very active," "active," "interested but non-active," or "not interested," creating a scale rank ordered from most active to most inactive.

= Interval Data =

Sometimes, in addition to being ordered, the differences (or intervals) between any two adjacent values on a measurement scale are the same. For example, the difference in temperature between 20 meters and 2 meters is the same as that between 90 meters and 91 meters. When each interval represents the same increment of the thing being measured, the measure is called an interval variable.

= Ratio Data =

Finally, in addition to having equal intervals, some measures also have an absolute zero point. That is, zero represents the absence of the thing being measured. Height and weight are obvious examples. Physicists sometimes use the Kelvin temperature scale, in which zero means the complete absence of energy. The same is not true of the Fahrenheit or Celsius (Centigrade) scales. Zero degrees Celsius, for example, represents the freezing point of water at sea level, but this does not mean that there is no temperature at this point. The choice to put zero degrees at this point on the scale is arbitrary. There is no particular reason why scientists could not have chosen instead the freezing point of beer in Rockhampton, Quensland (other than that water is a more common substance, at least for most successful scientists). With an absolute zero point, you can calculate ratios (hence the name). For example, $20 is twice as much as $10, but 60 degrees Fahrenheit is not really twice as hot as 30 degrees. Ratio data is fully quantitative: it tells us the amount of the variable being measured. The percentage of votes received by a candidate, Gross Domestic Product per Capita, and felonies per 100,000 population are all ratio variables.

= Dichotomous Variables =

Dichotomous variables (those with only two values) are a special case, and may sometimes be treated as nominal, ordinal, or interval. Take, for example, political party affiliation in a two-party legislature. Party is, on its face, a pure example of a nominal variable, with the values of the variable being simply the names of the parties (or arbitrary numbers used, for convenience, in place of the names). On the other hand, we could treat party (and other dichotomous variables) as ordinal, since there are only two possible ways for the values to be ordered, and it makes no difference which way is chosen. There is, therefore, no way that they can be listed out of order.

For certain purposes, we can even treat dichotomous variables as interval, since there is only one interval (the difference between Party A and Party B), which is obviously equal to itself.

= Why does it matter? =

The level of measurement makes a difference to the power of the statistical analysis that can be used. The higher the level of measurement of a variable, the more powerful are the statistical techniques that can be used to analyse it (note that "level of measurement" is itself an ordinal measure). For example, with nominal data, you can count the frequency with which each value of a variable occurs. A person's political voting, for example, is a nominal variable (with the values of the variable being "Liberal," "Labour," “Green," etc.), and so you can take data from a public opinion poll and count the number of respondents in the sample identifying with each party. You can also calculate each party's identifiers as a percentage of the sample total.  You can calculate joint frequencies and percentages (how many and what percent of Anglo-Australian are Liberal, for example).  You can also use certain measures that tell you how strong the overall relationship is between party and ethnicity, and the likelihood that the relationship occurred by chance.

On the other hand, there are other test you cannot legitimately perform with nominal data. Even if you use numbers to label candidates (e.g., 1 = Liberal, 2 = Labour, 3 = Green, etc.), you cannot really say that Liberal plus Labour equals Green, or that Green divided by Labour is half way between Liberal and Labour. In statistics, unfortunately, there are many techniques that require these kinds of assumptions. And for that, they require higher levels of measurement.

With ordinal data, you can employ techniques that take into account the fact that the values of a variable are listed in a meaningful order. With interval data, you can go even further and use powerful techniques that assume a measurement scale of equal intervals. As it happens, there are very few techniques in the social sciences that require ratio data, and so some textbooks ignore the distinction between interval and ratio scales.

If you use a technique that assumes a higher level of measurement than is appropriate for your data, you risk getting a meaningless answer. On the other hand, if you use a technique that fails to take advantage of a higher level of measurement, you may overlook important things about your data. (Note: in addition to level of measurement, many statistical techniques also require other assumptions about your data. For example, even if a variable is interval, some otherwise appropriate techniques may yield misleading results if the variable includes some values that are extremely high or low relative to the rest of the distribution.)

The distinctions between levels of measurement are not always simple or straightforward. Sometimes it depends on the underlying concept being measured. This applies, for example, to the question of whether to treat a dichotomous variable as nominal or ordinal. Do our values indicate two distinct categories (e.g., male and female), or do we think of them as two points along a spectrum (e.g., for or against offshore processing of asylum seekers; some people may favour or oppose offshore processing more strongly than others)?

In designing research, there can be tradeoffs between having data that are at a higher level of measurement and other considerations. Aggregate data (data about groups of people) are generally interval or ratio, but usually provide only indirect measures of how people think and act. Individual data get at these things more directly, but are usually only nominal or ordinal. Official election returns, for example, can provide us with ratio level data about the distribution of votes in each precinct. These data, however, tell us little about why individual people vote the way they do. Survey research (public opinion polling), which provides data that are for the most part only nominal or ordinal, allows us to explore such questions much more extensively and directly.

= Other ways to classify data = Sometimes you will find other terms used to describe the level of measurement of variables. SPSS, for example, distinguishes among nominal, ordinal, and scale (that is, interval or ratio) variables. Some texts distinguish between nonparametric (nominal or ordinal) and parametric (interval or ratio) variables. In describing different statistical procedures, we will sometimes distinguish between categorical and continuous variables. Categorical variables generally consist of a small number of values, or categories, and are usually nominal or ordinal. The values of continuous variables represent a large or even infinite number of possible points along a scale, and are interval or ratio.

= Further readings = Lane, David, et al. “Levels of Measurement,” Online Statistics: A Multimedia Course of Study

University of Cambridge. "Levels of Measurement," Universities' Collaboration in eLearning

=Credits = John L. Korey 2013, POLITICAL SCIENCE AS A SOCIAL SCIENCE, Introduction to Research Methods in Political Science: The POWERMUTT* Project,