**Observations **about **qualitative variables **are typically made in words but are possibly **coded **into **numbers **later on for purposes of **data processing**. **Observations **about **quantitative variables**, in contrast, are **numerical **at the outset. Anyone who works with **numbers**, therefore, must be very clear about what, if anything, is measured by them. A “3” that is a **code **word for a particular race or profession is a very different thing from a “3” that measures volume or weight. In fact, the assignment of **numbers** to characteristics that are being observed–which is **measurement**–can yield four types of **data **of increasing sophistication. It can produce **nominal**, **ordinal**, **interval**, or **ratio data**, and different **statistical concepts **and **techniques **are appropriately applied to each type.

# Nominal Data

The weakest level of **measurement**, producing the least amount of useful **information**, yields **nominal data**. These are **numbers **that merely name or label differences in kind and, thus, can serve the purpose of classifying **observations **about **qualitative variables **into mutually exclusive groups where the **numbers **in each group can then be counted. As noted, “male” might be coded as 0 and “female” as 1, but alternative labels of “male” = 100 and “female” = 17 would serve as well. Other examples of creating nominal data are classifying defective units of a product as 1 and satisfactory units as 2; labeling rooms on the first, second, or third floors by numbers in the 100s, 200s, or 300s, respectively; or designating rooms on the north or south side of a building by even or odd last digits. Then 102, 104, 106 would stand for first-floor rooms to the north; 301, 303, 305 for third-floor rooms facing south.

As these examples can confirm, it never makes sense to add, subtract, multiply, divide, rank, average, or otherwise manipulate **nominal data**, but one can count them. The presence of five 1s, in one of the above codes, denotes the presence of five females; 25 odd room numbers mean there are 25 rooms facing south. In contrast, adding eleven 0s and five 1s (because eleven men and five women are working in a firm) would yield a meaningless number 5. The fact that 1 is greater than 0 is equally meaningless: In what sense is “female” greater than “male”?

# Ordinal Data

The next level of **measurement **produces **ordinal data**. These are **numbers **that not only possess all the characteristics of **nominal data **but by their size also **order **or **rank observations **on the basis of importance. Differences between **numbers **or **ratios **of such **numbers**, however, remain meaningless. **Ordinal numbers **can be compared as greater than, smaller than, or equal to one another, but they contain no information about how much greater or smaller one number is compared to the other. Assessments of a product as superb, average, or poor might be recorded as 2, 1, 0, as 250, 10, 2, or even as 10, 9, 4.5–the important thing is that larger numbers denote a more favorable assessment, or a higher ranking, while smaller ones do the opposite. Yet these data make no statement about how much more or less favorable one assessment is compared to another. A 2 is deemed better than a 1 but not necessarily twice as good; a 250 is deemed better than a 10 but not necessarily 25 times as good; a 4.5 is deemed worse than 9 but not necessarily half as good; and that is all.

# Interval Data

Even more **information **is contained in **interval data**. These are **numbers **that possess all characteristics of **ordinal data **and, in addition, are related to one another by meaningful **intervals **or distances because all **numbers **are referenced to an admittedly arbitrary zero point. Given that arbitrariness, the **ratios **of such **numbers **are still meaningless. Addition and subtraction are permissible, but not multiplication and division. **Scales **of calendar time, clock time, and temperatures provide good examples of **measurements **that start from an arbitrarily located zero point and then utilize an equally arbitrary but consistent unit distance for expressing **intervals **between **numbers**. Consider how the Celsius scale places zero at the water-freezing point, whereas the Fahrenheit scale places it far below the freezing point. Within the context of either **scale**, the unit distance (degree of temperature) has a consistent meaning: each degree Celsius equals 1/100 of the distance between water’s freezing and boiling points; each degree Fahrenheit equals 1/180 of that distance. Note that the zero point, being arbitrarily located, does not denote the absence of the characteristic being measured. Unlike 0^{o} C nor 0^{o} F expresses a complete absence of heat. Note also that any **ratio **of **interval data **is meaningless; 90^{o} F is not twice as hot as 45^{o} F. Indeed, the ratio of the corresponding Celsius figures (32.2^{o} and 7.2^{o}) does not equal 2:1 but well over 4:1.

# Ratio Data

The highest level of **measurement**, producing the most useful **information**, yields **ratio data**. These are **numbers **that possess all the characteristics of **interval data **and, in addition, have meaningful **ratios **because they are referenced to an absolute or natural zero point that denotes the complete absence of the characteristic being measured. All types of **arithmetic operations**, even multiplication and division, can be performed with such data. Unlike the Fahrenheit-Celsius example, the ratio of any two such numbers is independent of the unit of measurement because each number is a distance measure from the same zero point. For example, the measurement of salaries, age, distance, height, volume, or weight produces ratio data. Because it is meaningful to rank salary data and to say that a salary of $15,000 is larger than one of $10,000, which is larger than one of $5,000, salary **data **give the kind of information provided by **ordinal data**. Yet, in addition, because it is meaningful to compare **intervals **between salary **data **and to say that the distance between $15,000 and $10,000 equals the distance between $10,000 and $5,000, these **data **also give the type of information provided by **interval data**. Further, these **data **are **ratio data **because we can safely describe $15,000 as three times as much money as $5,000; even a change in the unit of **measurement**, as from dollars to francs (at an exchange rate of, say, 6 francs to the dollar) does not change this conclusion: 1,500,000 cents still is three times as much money as 500,000 cents; 90,000 francs still is three times as much as 30,000 francs. Ultimately, this is true because zero dollars, zero francs, and zero cents all mean precisely the same thing. (In contrast, 0^{o} F ≠ 0^{o} C ≠ 0^{o} K.)

Variable | Associated Data Type | Description of Data | Permitted Operations | Examples |

Qualitative | Nominal | Numbers that merely name or label differences in kind | Placing numbers into mutually exclusive groups, counting numbers in each group | Highway or telephone numbers |

Qualitative | Ordinal | Numbers that not only label differences in kind, but that by their size also order or rank observations on the basis of importance, while differences between numbers or ratios of such numbers are meaningless | As above plus comparing numbers as greater than, smaller than, or equal to one another | The ranking of college courses as 3 for excellent, 2 for good, and 1 for poor; student class rank; the ranking of runners at the finish line |

Quantitative | Interval | Numbers that not only label differences in kind and by their size rank observations, but that are also referenced to an arbitrary zero point and between which intervals or differences are comparable, while their ratios are meaningless | As above plus adding and subtracting | Temperature scales; clock time; calendar time |

Quantitative | Ratio | Numbers that not only label differences in kind, by their size rank observations, and are separated by meaningful intervals, but that are also referenced to an absolute or natural zero point and between which ratios are meaningful as well | As above plus multiplying and dividing | Salary, measures of length, volume, weight |