In a classic chapter in his Handbook of Experimental Psychology, Stevens (1951) provided what is now the standard reference for the classification of types of numbers in measuring both quantitative and qualitative variables.

Stevens argues that there are four distinct ways in which we use numbers to measure phenomena: nominal, ordinal, interval, and ratio. In simple terms, we use numbers to

1. label or name items (nominal),
2. put things in order from largest to smallest or fastest to slowest or most beautiful to least beautiful (ordinal),
3. compare elements by determining the distance between them (interval), and
4. compare items by determining their relative, proportional characteristics (ratio).

In a nominal scale, numbers are used only as labels, and they carry no quantitative information beyond equality. That is, nominally scaled numbers identify different people, places, genders, and so on. For example, the numbers on the jersey of a hockey player say nothing about how skilled the player is. They only serve to tell the observer that player 22 is not equal to (is a different person from) player 25. This breaks down, for example, if players from two different teams have the same number, and we have to rely on jersey color in addition to the nominal number to determine inequality.

Another example of nominal scaling is your social security number. Actually, when these numbers were first issued, they did contain some ordinal information--they were issued in sequence. Nonetheless, the size of your social security number says nothing about your personal characteristics. It simply determines that you are a different person from your neighbor.

In an ordinal scale, the numbers represent not only different individuals, but also direction of difference. As people arrive for class, someone is first in the room, someone else second, another third, and so on. The numbers representing order of entry (1st, 2nd, 3rd...) tell us that there were different people entering the room, and that they came in at different times. However, they do not tell us how far apart the times of entry were. The first person might have been 5 minutes or 5 seconds ahead of the second person, who was 1 second or 2 hours ahead of the third, and so on.

Other examples of ordinally scaled variables include finishing position in a sports league or race; oldest, middle, and youngest children; and ranking of Fortune 500 companies according to investment risk.

Numbers in an interval scale represent different individuals, tell the direction of differences, and tell how large the differences are in units that remain constant across the scale. Scores on most psychological and educational tests fall in this category. The Scholastic Aptitude Test, for example, tells us that a person who scores 660 did 60 points better than a person who scored 600. However, we cannot say that the first person scored 10% better than the second person. That is because there is not true zero point on the SAT. You receive a score of 200 just for filling in your name correctly.

Consider a classroom test on arithmetic. After you have finished the course, you take an examination on which you receive the score of 0. Does that mean that you learned no arithmetic? Of course not. A zero on a classroom test does not mean the total absence of a quantity. Consequently, even in interval scales that contain a zero value, proportional or percentage comparisons are impossible and inappropriate.

Other examples of interval scales are temperature in the Fahrenheit or Celsius scales, where zero degrees does not indicate the total absence of heat; years of Julian calendar dating, in which there is no year 0000; and amount of energy in a physical system.

In a ratio scale, numbers represent different individuals, tell the direction of the differences, and tell how large the differences are in units which remain constant across the scale – all of the characteristics of nominal, ordinal, and interval scales. But in addition, due to the presence of an absolute zero that really represents none or the absence of the quantity being measured, ratio scales make it possible to compare proportions, percentages, and ratios.

Most measurements of physical quantity, like length, mass, volume, and duration of time are ratio. Percentage of games won and lost in baseball is also ratio, since it is based on counts of frequency, which are also ratio: Zero hams in the cupboard means you have no ham at all. Remember that it is the ratio scale that has a zero. A ratio scale is still ratio even if none of the actual scores are zero. Salary in a corporation is ratio, even though none of the employees earn $0.00.

Whether a set of numbers is measured on a nominal, ordinal, interval, or ratio scale influences what statistics are appropriate to deal with it. The mean of the jersey numbers of the players on the Buffalo Bills carries little significance. What information would be conveyed if we learned that the mean of the Bills players' numbers is higher than that of the Giants or the Dolphins? The mode is much more appropriate for football numbers--it might be interesting to discover that the favorite number among players is 12, for example. Favorite here is based on "most frequently occurring or chosen", thus, it is a mode.

For ordinal numbers, the mode is generally appropriate (the mode order of finish by American competitors in Olympic competition was 4), and the median may also be employed.

For interval and ratio scales, you may generally use any of the three measures of central tendency. Similar rules apply for measures of variability: Range only for nominal and ordinal scales, and range, variance, and standard deviation for interval and ratio scales.

The restriction of statistical analysis of different scales of numbers is a rational decision. Mathematically, numbers are numbers, and formulas and computers will compute any statistic on any set of numbers, regardless of the scale of measurement. Inappropriately applied, however, statistics may be misleading.

Comprehension exercise

For each of the following variables, define whether the scaling is nominal, ordinal, interval, or ratio. Then name the most appropriate measure(s) of central tendency and of variability for each variable.

Numbers on hockey players’ jerseys________________________________________

Social Security Numbers_________________________________________________

Order of students’ arrival in class__________________________________________

Order of finishers in the Boston Marathon___________________________________

SAT scores___________________________________________________________

Scores on a statistics quiz________________________________________________

Temperature in degrees Celsius____________________________________________

Temperature in degrees Kelvin_____________________________________________

Weight of Phramous widgets_______________________________________________

Duration of statistics classes_________________________________________________

Number of students who are tired in each class_________________________________

Salaries of Phramous executives_____________________________________________

Ages of automobiles______________________________________________________

Ages of Americans (Careful!)_______________________________________________

U.S. News rankings of the colleges___________________________________________

Valedictorian and salutatorian_______________________________________________

Speed of trucks on Interstate 90______________________________________________