Psychologist Stanley Smith Stevens proposed a taxonomy of *Levels
of Measurement*

- Nominal scale
- Ordinal scale
- Interval scale
- Ratio scale

*Stevens, S.S. (1946). On the Theory of Scales of Measurement.
Science, 103 2684, 677-80.*

- Qualitative
- Assigning names to things
- Group similar things together
- Taxonomies
- Ontologies

**Mathematical operations:**test for equality, test for set membership**Central tendency:**mode

- Semi qualitative, Rank
- Example: ‘completely agree’, ‘mostly agree’, ‘indifferent’, ‘mostly disagree’, ‘completely disagree’

- There is order, but intervals can be different
- Real differences between adjacent ranks may not be equal

**Mathematical operations:**less-than, sort**Central tendency:**median- And all the properties of
*Nominal*

- degree of difference between items, but not the ratio between them
- Examples
- Temperature in the Celsius scale
- date when measured from an arbitrary epoch (such as AD)
- location in Cartesian coordinates
- direction measured in degrees from north

**Mathematical operations:**Plus, Minus, ratio of differences**Central tendency:**arithmetic mean, standard deviation

- Has a meaningful (unique and non-arbitrary) zero value
- Most measurements in physical sciences and engineering
- mass, length, duration, plane angle, energy, and electric charge
- values have units of measurement

**Mathematical operations:**multiplication, division**Central tendency:**geometric mean, harmonic mean, coefficient of variation

- Names
- Grades (ordered labels like beginner, intermediate, advanced)
- Ranks (orders with 1 being the smallest or largest, 2 the next smallest or largest, and so on)
- Counted fractions (bound by 0 and 1)
- Counts (non-negative integers)
- Amounts (non-negative real numbers)
- Balances (any real number)

- Nominal
- Gradation of membership
- Ordinal
- Interval
- Log-interval
- Extensive ratio
- Cyclical ratio
- Derived ratio
- Counts
- Absolute

Incremental progress | Measure property | Mathematical operators | Advanced operations | Central tendency |
---|---|---|---|---|

Nominal | Classification, membership | =, ≠ | Grouping | Mode |

Ordinal | Comparison, level | >, < | Sorting | Median |

Interval | Difference, affinity | +, − | Yardstick | Mean, Deviation |

Ratio | Magnitude, amount | ×, / | Ratio | Geometric mean, Coefficient of variation |

Wikipedia: Levels of Measurement

These categories are used for teaching and education research

- Knowledge
- Comprehension
- Application
- Analysis
- Synthesis
- Evaluation

**Knowledge**“involves the recall of specifics and universals, the recall of methods and processes, or the recall of a pattern, structure, or setting.”**Comprehension**“refers to a type of understanding or apprehension such that the individual knows what is being communicated and can make use of the material or idea being communicated without necessarily relating it to other material or seeing its fullest implications.”

**Application**refers to the “use of abstractions in particular and concrete situations.”**Analysis**represents the “breakdown of a communication into its constituent elements or parts such that the relative hierarchy of ideas is made clear and/or the relations between ideas expressed are made explicit.”

**Synthesis**involves the “putting together of elements and parts so as to form a whole.”**Evaluation**engenders “judgments about the value of material and methods for given purposes.”

- Remember
- Understand
- Apply
- Analyze
- Evaluate
- Create

- Recognizing
- Recalling

- Interpreting
- Exemplifying
- Classifying
- Inferring
- Comparing
- Summarizing,
**Explaining**

- Executing
- Implementing

- Differentiating
- Organizing
- Attributing

- Checking
- Critiquing

- Generating
- Planning
- Producing