1.6 Data and Scales/Levels of Measurement

The following are the four basic scales/levels of data:

• Nominal

• Ordinal

• Interval

• Ratio

While each might produce the kinds of data you would see in a spreadsheet, the numbers may mean very different things, depending on the kind of data that was entered.

Nominal Data

For nominal data, the numbers assigned to the data have no inherent meaning. For example, if you assigned a 1 to denote a male and a 2 to denote a female in your spreadsheet, the numbers 1 and 2 do not stand for mathematical values. A woman is not mathematically twice a man! The numbers are simply labels for ease of entering data and for later analysis. It's also easier to type a 1 or 2 than it is to write "male" and "female" every time your data require it. You could just as easily label females 1 and males 2. Nothing about the data would change, but you would need to let the analysts know what the labels indicate.

Ordinal Data

For ordinal data, we know that the numbers indicate a change in magnitude, but the scale does not accurately indicate the direction or strength of differences. For example, if we know someone is in first place and someone else is in second, we know that the first person did something better, but none of the details. The numbers indicate a change in magnitude, but not one that we'd use math to make sense of. We certainly could not take the third-ranked score, divide by 3, and expect to see the first-ranked person's score or add the first- and second-ranked scores and expect them to add to the third-place score.

Similarly, even though you know the rank of each person, you do not know how much better they did, only that there was a difference. As an example, we might usually find a small difference in the marathon times between the first- and second-place runners but a very large difference between the last runner and the penultimate runner who arrived before them. For ordinal data, it only matters what place each runner finished in and not how far ahead of the next runner they were.

Interval Data

Interval data shares a property of ordinal data in that numbers mean a change in magnitude, but for interval data each unit of difference is the same, no matter where it might occur on the scale. On an interval scale, 1 and 2 are just (and only) as different from each other as 101 and 102 are. Temperature is a classic example of interval data. In degrees Celsius, the difference between 60 and 61 degrees is the same difference as that between 30 and 31 degrees, because the interval between units is always the same.

An interesting question arises, though, when you ask if 60 degrees Celsius is twice as hot as 30 degrees Celsius. While the number 60 is double that of 30, is 60 degrees actually twice as hot as 30 degrees? Imagine someone (call him Andrew) who is the average height of males in the United States at 5 feet 10 inches. We'll call this height 0 Andrew units (0A) and measure our intervals in inches. Thus, someone who is 1A is 5'11'' and someone is 2A is 5'12'', or 6'. While it is true that 2 is double 1, someone who is 2A on our scale is not twice as tall as someone who is 1A, because 0A does not actually mean a person has no height. Looking back to the Celsius scale, while 0°C means freezing, it does not actually mean an absence of heat (which would be absolute zero, or -273.15°C). So when we think of a true zero, 30°C is actually 303.15 heat units and 60°C is 333.15 heat units, which is nowhere near double. Why is this important? We still cannot use many mathematical processes (addition, subtraction, multiplication, etc.) to make sense of our data.

Ratio Data

The advantage of a true zero is that numbers can (finally!) be added, subtracted, multiplied, divided, integrated, and so on and still have real meaning. This is the hallmark of the kind of data known as ratio data. While analyses can be performed on each kind of data, ratio data is sometimes the easiest to understand and compare. As an example, we can use ratio data to compare two companies, one who makes billions of dollars and one who makes around $100,000. A ratio would allow us to compare the companies' efficiencies, even though their profit margins are vastly different.