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.

“Giving names to things”

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

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

Can you name examples?

- 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*

Can you give more examples?

- Difference between items, but not the ratio between them
- Temperature in the Celsius scale
- date 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

What is the meaning of “Twice the temperature”?

- 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

Examples?

- 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

*In my first day of classes, the student’s union gave us a
newspaper with this story in the first page:*

The following concerns a question in a physics degree exam at the University of Copenhagen:

“How could you measure the height of a tall building, using a barometer?”

“You tie a long piece of string to the neck of the barometer, then lower the barometer from the roof to the ground.

The length of the string plus the length of the barometer will equal the height of the building.”

The professor said: *“the answer is technically correct but does
not demonstrate a knowledge of physics”*

“Take the barometer up to the roof, drop it over the edge, and measure the time it takes to reach the ground.

The height of the building can then be worked out from the formula \(h = gt^2/2\)”

The professor said: *“the answer is technically correct but does
not demonstrate a knowledge of physics”*

“If the sun is shining you could measure the height of the barometer, then set it on end and measure the length of its shadow.

Then you measure the length of the building’s shadow, and thereafter it is a simple matter of proportional arithmetic to work out the height.”

The professor said: *“the answer is technically correct but does
not demonstrate a knowledge of physics”*

“But if you wanted to be highly scientific about it, you could tie a short piece of string to the barometer and swing it like a pendulum, first at ground level and then on the roof of the skyscraper.

The height is worked out by the difference in the gravitational restoring force \(T =2π\sqrt{l /g}\).”

*“the answer is technically correct but does
not demonstrate a knowledge of physics”*

“Tying a piece of string to the barometer, which is as long as the height of the building, and swinging it like a pendulum, and from the swing period, calculate the pendulum length.”

*“the answer is technically correct but does
not demonstrate a knowledge of physics”*

“Or if the skyscraper has an outside emergency staircase, it would be easier to walk up it and mark off the height of the skyscraper in barometer lengths, then add them up.”

*“the answer is technically correct but does
not demonstrate a knowledge of physics”*

“I would go to the building superintendent and offer him a brand-new barometer if he will tell me the height of the building!”

*“the answer is technically correct but does
not demonstrate a knowledge of physics”*

Measuring the pressure difference between ground and roof and calculating the height difference

This is how planes measure altitude

Air pressure reduces when we go up

The same attribute can be measured through different physical properties

Thinking “out of the box” enables measuring things that were not easy to measure before