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Subject 2. Measurement Scales

Measurement is the assignment of numbers to objects or events in a systematic fashion. To choose the appropriate statistical methods for summarizing and analyzing data, we need to distinguish between different measurement scales or levels of measurement.

  • Nominal Scale

    • Nominal measurement represents the weakest level of measurement.
    • It consists of assigning items to groups or categories.
    • No quantitative information is conveyed and no ordering (ranking) of the items is implied.
    • Nominal scales are qualitative rather than quantitative.

    Religious preference, race, and sex are all examples of nominal scales. Another example is portfolio managers categorized as value or growth style will have a scale of 1 for value and 2 for growth. Frequency distributions are usually used to analyze data measured on a nominal scale. The main statistic computed is the mode. Variables measured on a nominal scale are often referred to as categorical or qualitative variables.

  • Ordinal Scale

    • Measurements on an ordinal scale are categorized.
    • The various measurements are then ranked in their categories.
    • Measurements with ordinal scales are ordered with higher numbers representing higher values. The intervals between the numbers are not necessarily equal.

    Example 1

    On a 5-point rating scale measuring attitudes toward gun control, the difference between a rating of 2 and a rating of 3 may not represent the same difference as that between a rating of 4 and a rating of 5.

    Example 2

    Two categories might be value and growth. Within each category, the portfolio managers measured will be weighted according to performance on a scale from 1 to 10, with 1 being the best- and 10 the worst-performing manager.

    There is no "true" zero point for ordinal scales, since the zero point is chosen arbitrarily. The lowest point on the rating scale in the example was arbitrarily chosen to be 1. It could just as well have been 0 or -5.

  • Interval Scale

    Interval scales rank measurements and ensure that the intervals between the rankings are equal. Scale values can be added and subtracted from each other.

    For example, if anxiety was measured on an interval scale, a difference between a score of 10 and a score of 11 would represent the same difference in anxiety as the difference between a score of 50 and a score of 51.

    Interval scales do not have a "true" zero point. Therefore, it is not possible to make statements about how many times higher one score is than another. For the anxiety example, it would not be valid to say that a person with a score of 30 was twice as anxious as a person with a score of 15. True interval measurement is somewhere between rare and nonexistent in the behavioral sciences. No interval scales measuring anxiety, such as the one described in the example, actually exist. A good example of an interval scale is the Fahrenheit measure of temperature. Equal differences on this scale represent equal differences in temperature, but a temperature of 30°F is not twice as warm as one of 15°F.

  • Ratio Scale

    Ratio scales are like interval scales except that they have true zero points. This is the strongest measurement scale. In addition to permitting ranking and addition or subtraction, ratio scales allow computation of meaningful ratios. A good example is the Kelvin scale of temperature. This scale has an absolute zero. Thus, a temperature of 300°K is twice as high as a temperature of 150°K. Two financial examples of ratio scales are rates of return and money. Both examples can be measured on a zero scale, where zero represents no return, or in the case of money, no money.

Note that as you move down through this list, the measurement scales get stronger.

Hint: Remember the order of the different scales by remembering NOIR (the French word for black); the first letter of each word in the scale is indicated by the letters in the word NOIR.

Before we move on, here's a quick exercise to make sure that you understand the different measurement scales. In each case, identify whether you think the data is nominal, ordinal, interval or ratio:

  • The number of goals scored by a soccer player in a season.
  • The temperature in Fahrenheit.
  • The relative positions of contestants in a beauty pageant.
  • The speed at which a vehicle travels.
  • The allocation of the number "1" to boys and "2" to girls in a class.

Answers and Explanations

  • Ratio: numbers have meaning and can't go below 0.
  • Interval: temperature is always measured on the interval scale.
  • Ordinal: relative positions are ranks, so this data has been ranked.
  • Ratio: speed is a meaningful number and cannot be negative.
  • Nominal: the numbers are labels, and have no other meaning.

Practice Question 1

The following chart illustrates the breakdown of a survey of 300 bachelors, living in Florida, as to their favorite fast food restaurant.

Favorite Fast Food Restaurant
Burger King ================= 51
Hardee's =============== 45
McDonald's ====================== 66
Taco Bell ========== 30
Wendy's ============================ 84
Other ======== 24

What level of data is the number of bachelors?

A. ratio
B. nominal
C. ordinal

Correct Answer: A

Practice Question 2

A student's score on the ACT test would be what level of measurement?

A. ordinal
B. interval
C. ratio

Correct Answer: B

Practice Question 3

If thirty students listed the distance that they drive to school, this data would be what level of measurement?

A. ordinal
B. interval
C. ratio

Correct Answer: C

Practice Question 4

If an administrator had someone count the number of people in each room of an office building, this data would be what level of measurement?

A. nominal
B. interval
C. ratio

Correct Answer: C

Practice Question 5

The years in which the U.S. gross domestic product decreased would be an example of which level of measurement?

A. ratio
B. interval
C. ordinal

Correct Answer: B

The correct answer is interval because it makes sense to talk about differences between years (1984 - 1963 = 21 years); ratios are meaningless since time did not begin in year 0.

Study notes from a previous year's CFA exam:

2. Measurement Scales