Basic Concepts of Measurement (5 min.)

• Variable
• Construct
• Values & Scores
• Measurement & Operational Definition
• Comprehensive Example Summarizing Above Concepts
• Measurement Scale: Nominal (Categorical), Ordinal (Rank-Order), (Equal-) Interval, and Ratio

Variable

• Variable: Any characteristic of people or things you (researchers) are interested in. For example, height or intelligence of people.
  • [Star Q] “Why is it called a variable? Because it …”
  • [Star A] “Varies”
  • Variable is a characteristic that varies across people or situation. Therefore, it is called a variable.
    • Usually, we are not interested in something that does not vary (everybody has only one brain or two eyes and these facts are not interesting ones).
    • [Star Q] “What do we call something that does not vary across people or situation?”
    • [Star A] “A Constant”
    • [Star Q] “What are the examples of variables?”

[Star A] Examples of Variable (Things that vary across people or situations)

Gender, Job

Birth-order, USA Olympic rank

Body temperature, IQ

Hours of activity (work, sleep, study, cell phone use), Number of texts/calls

• [Star Q] “Can you think of a variable of things rather than human-related variables?”
• [Star A] “Cars’ color or gas mileage”

Construct

• Construct: Some variables that are complex to measure because they are multi-dimensional and cannot be observed directly. For example, one’s tendencies or thought processes like anxiety or intelligence are difficulty measure.
  • [Star Q] “Can you think of any examples of a construct of things?”
  • [Star A] “Qualities of movies”

Values and Scores

• Values: All the possible numbers/names a variable or construct can have. For example, numbers from 1 to 10 that are used to measure one’s pain (1 being No Hurt; 10 being Hurts Worst). As another example, the words “dull or acute” that a patient uses to describe her pain. All those numbers or words that could have been used to refer to a specific level of the variable are called values.
• Scores: A specific score an individual gets. If Henry describes his pain as 2 or dull, then these are the scores of Henry for his pain variable.
  • [Star Q] “What is the difference between the values and scores?”
  • [Star A] “Values are all the possible numbers or names while a score is a specific value of an individual”

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Measurement and Operational Definition

• Measurement: Assignment of scores to people or things to represent the individual’s variable. For example, to measure a kid’s height, one can use a tape measure and assign a specific score as the height of the child.
• Operational Definition: Measurement involves a set of procedures or operations such as reading the numbers of a tape measure. In science, those procecures or operations themselves can often be used as a definition of the variable. For example, the definition of one’s height is the number read from a tape measure that juxtaposed to the person. As another example, depression can be defined as a set of scores obtained from a psychological test designed to measure the depression (Beck’s Depression Scale).
  • [Star Q] “Why would we want to use operational definition to define a variable instead of using one’s own definition for the variable?”
  • [Star A] “To form a common ground for an objective communication regarding the variable”

Measurement Scale: Nominal (Categorical), Ordinal (Rank-Order), (Equal-) Interval, and Ratio

Finally, we will talk about the four scales of measurement:

1. Nominal (Categorical)
2. Ordinal or Rank Order
3. Interval or Equal Interval
4. Ratio

To discuss this topic, we should first read the following passage from Saint Exupery’s novel, The Little Prince:

Many scientists including psychologists are grown-ups who like numbers. They often give numbers to the values of things (variables) they are interested in and measure. For example, one’s age is measured in terms of the years the person has lived according to the information on an id or birth certificate and how much one weighs is measured using a bathroom scale. They do so (assign numbers) not only to quantitative things (How much does his father make? e.g., $100,000 per year) but also to qualitative things (Is the person female or male? 1 being female and 2 being male).

Depending on the nature of the numeric values of a variable, the variable can be categorized into one of four categories.

I always ask my students to think about the 1, 2, and 3 of a variable. Depending on the variable, the 1, 2, and 3 could mean:

Age: 1 year old, 2 years old, 3 years old,

Number of siblings: 1 brother, 2 brothers, 3 brothers

Weight: 50 pound, 100 pound, 150 pound

If you want to know whether or not the variable is a nominal, ordinal, interval, and ratio, you need to consider the 1, 2, and 3 of the variable and follow the decision-making process as below:

1. Nominal (Categorical) variable: If the values 1, 2, 3 (and so on) of a variable merely refer to different categories/members of the variable (e.g., 1 being female and 2 being make) while the magnitude of the numbers does not refer to different degree of a certain characteristic, then, it is a nominal (categorical) variable.. That is, the magnitude of the values of the variable is meaningless as different numbers arbitrarily refer to different things. For example, gender, color, race are nominal variables. Specifically, when scientists assign numbers to gender, color, race, the magnitudes are meaningless and different numbers merely refer to different gender, color, and race.
2. Ordinal (Rank-Order) variable: If the values 1, 2, 3 (and so on) of a variable not only refer to different categories/members of the variable but also systematically refer to different levels/degrees of the variable (e.g., 1 is being better/worse than 2, and 2 is being better/worse than 3, and so on), then, it is an ordinal (rank-order) variable. For example, rankings of countries in Olympic not only refer to different countries (i.e., nominal) but also refer to different levels of sports abilities. Therefore, ranking is an ordinal variable. However, one cannot assume that the distance between 1 and 2 and the distance between 2 and 3 are equal. For example, United States, China, and Russia are the top three countries in Tokyo 2020 Olympic. However, we cannot assume that the sports-ability difference between US and China is exactly the same as the difference between China and Russia. Therefore, the different ranks merely refer to different levels/degrees of the variable (sports ability) without assuming equal distance across values.
3. Interval (Equal-Interval) variable: If the values 1, 2, 3 (and so on) of a variable have both characteristics mentioned above (different values refer to different categories/members and the magnitude of the values systematically refer to different levels/degrees of the variable) and one can also assume that that the distance between 1 and 2 is equal to the distance between 2 and 3, then, it is an interval (equal-interval) variable. Room temperature is a typical example of an interval (equal-interval) variable. That is, 1, 2, 3 degrees in either Celsius or Fahrenheit are not only refer to different degrees (nominal) but also systematically refer to increasing temperature (ordinal). In addition, one can assume that the temperature difference between 1 and 2 is exactly the same as the temperature difference between 2 and 3. However, 0 degree does not mean absolute zero/nothing of the variable, which is a required characteristic of a ratio variable (see below).
4. Ratio variable: If the values 1, 2, 3 (and so on) of a variable have all the above characteristics and the value 0 of the variable (if there is any) refers to absolute zero/nothing of a variable, then, the variable is a ratio variable. Many physical measurement (e.g., height or weight) are ratio variables. Ratio variable also allows one to say that value 4 of a variable is two times greater than the value 2 of the same variable. For example, we can say that 10 feet is two times longer than 5 feet. *However, in case of an interval variable (without absolute 0), the lack of absolute 0 prevents one from making a ratio calculation. For example, one cannot say that 90 degrees is two times hotter than 45 degrees.

Scale	Variables
Nominal (Categorical)	Gender, Whether or not taking a previous stat class (yes/no), Dream job
Ordinal (Rank-Order)	Birth-order, USA Olympic rank, Rank-ordered criteria of Partner choice
Interval (Equal-Interval)	Room temperature / IQ Scores
Ratio	Hours of (work, sleep, study, cell phone), number of texts, calls, laughs

Comprehensive Example Summarizing Above Concepts

• Movie critic website Rotten Tomatoes has their own movie-quality evaluation system called Tomatometer (The term was created based on the fact that audience used to express their feeling after watching a show by throwing things including vegetables).
• They want to measure the movie quality (variable, which is also a construct as it is a complex and difficult-to-measure variable) using the following values (there are three values):
  1. A fresh red tomato: When at least 60% of reviews for a movie or TV show are positive, a red tomato is displayed to indicate its Fresh status.
  2. A green splat tomato: When less than 60% of reviews for a movie or TV show are positive, a green splat is displayed to indicate its Rotten status.
  3. Tomato with banners reads “Certified Fresh”: Certified Fresh status is a special distinction awarded to the best-reviewed movies and TV shows. To qualify, movies or TV shows must meet some requirements including “A consistent Tomatometer score of 75% or higher,” and so on.
• A movie can have any of these values as its specific score.
• Providing such a specific score to a movie is called measurement.
• The specific procedure or operation described next to each value (recruit reviewers and make them rate, then, see what % of them gave positive reviews) that they use to determine a specific score of a movie is called the operational definition of the variable, movie quality.
• The three different values not only refer to different movies but also different quality in a systematic way (a tomato with the banner is the best, followed by a fresh red tomato, which is better than a green splat tomato). However, one cannot assume that the movie-quality difference between A tomato with banner and a fresh red tomato is the same to the difference between a fresh red tomato and a green splat tomato, the Tomatometer is an ordinal (rank-order) scale.