**Levels of Measurement**

In 1946, Harvard University psychologist Stanley Smith Stevens developed the theory of the four levels of measurement when he published an article in *Science* entitled, "On the Theory of Scales of Measurement." In this famous article, Stevens argued that all measurement is conducted using four measurement levels. The four levels of measure, in order of complexity, are:

**N**ominal

**O**rdinal

**I**nterval

**R**atio

Here is a simple trick for remembering the four levels of measurement: Think "**NOIR**." Noir is the French word for black. "N" is for nominal. "O" is for Ordinal. "I" is for Interval. And, "R" is for ratio.

**Categorical and Quantitative Measures:**

The nominal and ordinal levels are considered categorical measures while the interval and ratio levels are viewed as quantitative measures.

**Knowing the level of measurement of your data is critically important** as the techniques used to display, summarize, and analyze the data depend on their level of measurement.

Let us turn to each of the four levels of measurement.

**A. The Nominal Level**

The nominal level of measurement is the simplest level. "Nominal" means "**existing in name only**." With the nominal level of measurement all we can do is to **name** or **label** things. Even when we use numbers, these numbers are only names. We cannot perform any arithmetic with nominal level data. All we can do is count the frequencies with which the things occur.

With nominal level of measurement, no meaningful order is implied. This means we can re-order our list of variables without affecting how we look at the relationship among these variables.

Here are some examples of nominal level data:

- The number on an athlete's uniform
- Your social security number
- Your Visa card number
- Your political party affiliation
- The city where you were born
- Your religion
- Your social security number
- The color of your eyes
- The color of your hair
- The color of the candies in a bag of M&Ms

With the nominal level of measurement, we are limited in the types of analyses we can perform. We can count the frequencies of items of interest, but we cannot sort the data in a way that changes the relationship among the variables under investigation. We can calculate the mode of the frequently occurring value or values. And, we can also perform a variety of non-parametric hypotheses tests. Non-parametric tests make no assumptions regarding the population from which the data are drawn. But, we cannot calculate common statistical measures like the mean, median, variance, or standard deviation.

**B. The Ordinal Level**

The ordinal level of measurement is a more sophisticated scale than the nominal level. This scale enables us to order the items of interest using ordinal numbers. Ordinal numbers denote an item's position or rank in a sequence: First, second, third, and so on. But, we lack a measurement of the distance, or intervals, between ranks. For example, let's say we observed a horse race. The order of finish is Rosebud #1, Sea Biscuit #2, and Kappa Gamma #3. We lack information about the difference in time or distance that separated the horses as they crossed the finish line.

Here are some examples of ordinal level data:

- Order of finish in a race or a contest
- Letter grades: A, B, C, D, or F
- Ranking of chili peppers on a scale of hot, hotter, hottest
- A student's year of study in high school or college: Freshman, Sophomore, Junior, and Senior
- Stage of cancer: Stage I, II, III, or IV
- Level of agreement: Strongly Disagree, Disagree, Neutral, Agree, Strongly Agree

With the ordinal level of measurement, we can count the frequencies of items of interest and sort them in a meaningful rank order. And, as we said, we cannot, however, measure the distance between ranks. In terms of statistical analyses, we can count the frequency of an occurrence of an event, calculate the median, percentile, decile, and quartiles. We can also perform a variety of non-parametric hypotheses tests. But, we cannot calculate common statistical measures like the mean, median, variance, or standard deviation. And, we cannot perform parametric hypothesis tests using z values, t values, and F values.

**C. The Interval Level**

With the interval level of measurement we have quantitative data. Like the ordinal level, the interval level has an inherent order. But, unlike the ordinal level, we do have the distance between intervals on the scale. The interval level, however, lacks a real, non-arbitrary zero.

To repeat, here are three characteristics of the interval level:

- The values have a meaningful order
- The distances between the ranks are measureable
- There is no "true" or natural zero

The classic example of the interval scale is temperature measured on the Fahrenheit or Celsius scales. Let's suppose today's high temperature is 60º F and thirty days ago the high temperature was only 30º F. We can say that the difference between the high temperatures on these two days is 30 degrees. But, because our measurement scale lacks a real, non-arbitrary zero, we cannot say the temperature today is twice as warm as the temperature thirty days ago.

In addition to temperature on the Fahrenheit or Celsius scales, examples of interval scale measures include:

- Scores on the College Board's Scholastic Aptitude Test, which measures a student's scores on reading, writing, and math on a scale of 200 to 800
- Intelligence Quotient scores
- Dates on a calendar
- The heights of waves in the ocean
- Longitudes on a globe or map
- Shoe size

With the interval level of measurement, we can perform most arithmetic operations. We can calculate common statistical measures like the mean, median, variance, or standard deviation. But, because we lack a non-arbitrary zero, we cannot calculate proportions, ratios, percentages, and fractions. We can also perform all manner of hypotheses tests as well as basic correlation and regression analyses.

**D. The Ratio Level**

The last and most sophisticated level of measurement is the ratio level. As with the ordinal and interval levels, the data have an inherent order. And, like the interval level, we can measure the intervals between the ranks with a measurable scale of values. But, unlike the interval level, we now have meaningful zero. The addition of a non-arbitrary zero allows use to calculate the numerical relationship between values using ratios: fractions, proportions, and percentages.

An example of the ratio level of measurement is weight. A person who weights 150 pounds, weights twice as much as a person who weighs only 75 pounds and half as much as a person who weighs 300 pounds. We can calculate ratios like these because the scale for weight in pounds starts at zero pounds.

n addition to weight, examples of ratio scale measures include:

- Height
- Income
- Distance travelled
- Time elapsed or time remaining
- Money in your bank account, wallet, or pocket

With the ratio level of measurement, we can perform all arithmetic operations including proportions, ratios, percentages, and fractions. In terms of statistical analyses, we can calculate the mean, geometric mean, harmonic mean, median, mode, variance, and standard deviation. We can also perform all manner of hypotheses tests as well as correlation and regression analyses.

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