1.2. Levels of measurement¶

a.k.a. Scales of measure
Developed by Stanley Smith Stevens

Measurements in science use following scales

Nominal scale
Ordinal scale
Interval scale
Ratio scale

Examples of different scale types

Scale type	Example
Nominal	Rocks as igneous, sedimentary, metamorphic; first names
Ordinal	Rank-ordering data: Result of horse race, Mohs scale of mineral hardness
Interval	temperature with the Celsius scale,
Ratio	Mass, length, time, plane angle, energy and electric charge

Mathematical features of different scale types

Scale type	Permissible statistics	Admissible scale transformation	Mathematical structure
Nominal	Mode,Chi-squared	One-to-one (equality)	Standard set (unordered)
Ordinal	Median,percentile	Monotonically increasing (order <)	Totally ordered set
Interval	Mean,std,correlation,regression	Positive linear (affine)	Affine line
Ratio	All above+geometric mean,log etc	Positive similarities	1-D vector space

We can notice, that understanding of scale is important in appropriate choice of statistical measures for data analysis.

1.2.1. Nominal scale¶

Nominal measures offer names or labels for certain characteristics
Objects are classified using labels
- Rocks as igneous, sedimentary, metamorphic
- A group of people classified based on their first name
Valid operations
- Equivalence
- Set memberships

1.2.1.1. Categorical variables¶

Variables assessed on a nominal scale are called categorical variables.
Binary variables (or Bernoulli variables) : only two possible categories
- Yes or no
- success or failure
Multi-way variables

1.2.1.2. Categorical distribution¶

A categorical distribution is a probability distribution that describes the result of a random event that can take one of K possible outcomes, with the probability of each outcome separately specified.

There is not necessarily an underlying ordering of these outcomes.
Numerical labels are attached for convenience.

1.2.1.3. Statistics¶

Central tendency is given by its mode [e.g. most common name]
Mean is not defined [what would be average name in a set of people?]
Median is not defined [There is no ordering in the labels]

1.2.2. Ordinal scale¶

Rank ordering data puts the data on ordinal scale
Order of measurements is described.
Relative size or degree of difference between measured items is not described.

Examples

Result of a horse race, where the horses are ordered based on which one arrived 1st, second, or third, etc..
Names arranged in alphabetical order
- We can say which name comes first which later in this order.
- But there is no meaning of difference between names.
Psychometric measurements [like IQ etc.]
Food quality : exceptional, great, good, average, bad, poor

1.2.2.1. Statistics¶

Central tendency specified using mean or median
Mean cannot be defined

1.2.2.2. Order isomorphism¶

An ordinal scale defines a total preorder of objects
Scale values may be sorted on a single line with no ambiguities
Numbers may be assigned to scale values
Any transformation of numbers using a monotonically increasing function doesn’t change the order, hence retains validity.
This is known as order isomorphism.

1.2.3. Interval scale¶

Quantitative attributes are measurable on interval scale
Difference between levels is meaningful.
Difference can be multiplied to exceed or equal another difference.
Ratio between numbers of this scale is not meaningful
Multiplication and division cannot be done directly.
Ratio between differences is meaningful.
Choice or origin is arbitrary and not meaningful. [e.g. 0 degree Celsius]

1.2.3.1. Statistics¶

Central tendency can be represented by mode, median, mean all.
Statistical dispersion can be measured using, standard deviation, quantiles etc.
Studentized range or coefficient of variation is not supported.
Moments are not useful since origin is arbitrary. Central moments make sense.

1.2.4. Ratio scale¶

Measurement is the estimation of ratio between magnitude of a continuous quantity and a unit magnitude of same kind.
A zero value is supported.