# 1.4: Levels of Measurement

- Page ID
- 5694

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Levels of Measurement

Some researchers and social scientists use a more detailed distinction of measurement, called the **levels of measurement**, when examining the information that is collected for a variable. This widely accepted (though not universally used) theory was first proposed by the American psychologist Stanley Smith Stevens in 1946. According to Stevens’ theory, the four levels of measurement are nominal, ordinal, interval, and ratio.

Each of these four levels refers to the relationship between the values of the variable.

### Nominal measurement

A **nominal measurement** is one in which the values of the variable are names.

**Ordinal measurement**

An **ordinal measurement** involves collecting information of which the order is somehow significant. The name of this level is derived from the use of ordinal numbers for ranking (1st, 2nd, 3rd, etc.).

### Examples of Nominal and Ordinal Measurements

The names of the different species of Galapagos tortoises are an example of a nominal measurement.

If we measured the different species of tortoise from the largest population to the smallest, this would be an example of ordinal measurement. In ordinal measurement, the distance between two consecutive values does not have meaning. The 1st and 2nd largest tortoise populations by species may differ by a few thousand individuals, while the 7th and 8th may only differ by a few hundred.

**Interval measurement**

With **interval measurement**, there is significance to the distance between any two values.

**Ratio measurement**

A **ratio measurement** is the estimation of the ratio between a magnitude of a continuous quantity and a unit magnitude of the same kind. A variable measured at this level not only includes the concepts of order and interval, but also adds the idea of 'nothingness', or absolute zero.

### Examples of Interval and Ratio Measurement

We can use examples of temperature for these.

An example commonly cited for interval measurement is temperature (either degrees Celsius or degrees Fahrenheit). A change of 1 degree is the same if the temperature goes from 0∘ C to 1∘ C as it is when the temperature goes from 40∘ C to 41∘ C. In addition, there is meaning to the values between the ordinal numbers. That is, a half of a degree has meaning.

With the temperature scale of the previous example, 0∘ C is really an arbitrarily chosen number (the temperature at which water freezes) and does not represent the absence of temperature. As a result, the ratio between temperatures is relative, and 40∘ C, for example, is not twice as hot as 20∘ C. On the other hand, for the Galapagos tortoises, the idea of a species having a population of 0 individuals is all too real! As a result, the estimates of the populations are measured on a ratio level, and a species with a population of about 3,300 really is approximately three times as large as one with a population near 1,100.

### Comparing the Levels of Measurement

Using Stevens’ theory can help make distinctions in the type of data that the numerical/categorical classification could not. Let’s use an example from the previous section to help show how you could collect data at different levels of measurement from the same population.

### Determining Levels of Measurement

Assume your school wants to collect data about all the students in the school.

If we collect information about the students’ gender, race, political opinions, or the town or sub-division in which they live, we have a nominal measurement.

If we collect data about the students’ year in school, we are now ordering that data numerically (9th, 10th,11th, or 12th grade), and thus, we have an ordinal measurement.

If we gather data for students’ SAT math scores, we have an interval measurement. There is no absolute 0, as SAT scores are scaled. The ratio between two scores is also meaningless. A student who scored a 600 did not necessarily do twice as well as a student who scored a 300.

Data collected on a student’s age, height, weight, and grades will be measured on the ratio level, so we have a ratio measurement. In each of these cases, there is an absolute zero that has real meaning. Someone who is 18 years old is twice as old as a 9-year-old.

It is also helpful to think of the levels of measurement as building in complexity, from the most basic (nominal) to the most complex (ratio). Each higher level of measurement includes aspects of those before it. The diagram below is a useful way to visualize the different levels of measurement.

## Example

Use the approximate distribution of Giant Galapagos Tortoises in 2004 to answer the following questions.

Island or Volcano |
Species |
Climate Type |
Shell Shape |
Estimate of Total Population |
Population Density (per km2) |
Number of Individuals Repatriated∗ |
---|---|---|---|---|---|---|

Wolf | becki | semi-arid | intermediate | 1139 | 228 | 40 |

Darwin | microphyes | semi-arid | dome | 818 | 205 | 0 |

Alcedo | vanden- burghi | humid | dome | 6,320 | 799 | 0 |

Sierra Negra | guntheri | humid | flat | 694 | 122 | 286 |

Cerro Azul | vicina | humid | dome | 2.574 | 155 | 357 |

Santa Cruz | nigrita | humid | dome | 3,391 | 730 | 210 |

Española | hoodensis | arid | saddle | 869 | 200 | 1,293 |

San Cristóbal | chathamen- sis | semi-arid | dome | 1,824 | 559 | 55 |

Santiago | darwini | humid | intermediate | 1,165 | 124 | 498 |

Pinzón | ephippium | arid | saddle | 532 | 134 | 552 |

Pinta | abingdoni | arid | saddle | 1 | Does not apply | 0 |

### Example 1

What is the highest level of measurement that could be correctly applied to the variable 'Population Density'?

- Nominal
- Ordinal
- Interval
- Ratio

Population density it quantitative data, which means it will either fall into the nominal or ordinal categories. Now we just have to think about whether it has a true zero. Does a population density of 0 mean that there really is no population density? Yes, that is the correct meaning, so it is a true zero. This means that the highest level of measurement is ratio.

Note: If you are curious about the “does not apply” in the last row of Table 3, read on! There is only one known individual Pinta tortoise, and he lives at the Charles Darwin Research station. He is affectionately known as Lonesome George. He is probably well over 100 years old and will most likely signal the end of the species, as attempts to breed have been unsuccessful.

## Review

For 1-4, identify the level(s) at which each of these measurements has been collected.

- Lois surveys her classmates about their eating preferences by asking them to rank a list of foods from least favorite to most favorite.
- Lois collects similar data, but asks each student what her favorite thing to eat is.
- In math class, Noam collects data on the Celsius temperature of his cup of coffee over a period of several minutes.
- Noam collects the same data, only this time using degrees Kelvin.

For 5-8, explain whether or not the following statements are true.

- All ordinal measurements are also nominal.
- All interval measurements are also ordinal.
- All ratio measurements are also interval.
- Steven’s levels of measurement is the one theory of measurement that all researchers agree on.

For 9-11, indicate whether the variable is ordinal or not. If the variable is not ordinal, indicate its variable type.

- Opinion about a new law (favor or oppose)
- Letter grade in an English class (A, B, C, etc.)
- Student rating of teacher on a scale of 1 – 10.

For 12-14, explain whether the quantitative variable is continuous or not:

- Time it takes for student to get from home to school
- Number of hours a student studies per night
- Height (in inches)

- Give an example of an ordinal variable for which the average would make sense as a numerical summary.
- Find an example of a study in a magazine, newspaper or website. Determine what variables were measured and for each variable determine its type.
- How do we summarize, display, and compare data measured at different levels?

## Additional Resources

Practice for Levels of Measurement