# Scale of measurement

Stevens' Scales of Measurement or level of measurement is a system for classifying attribute data into four categories, developed by psychologist Stanley Smith Stevens and first published in 1946. Stevens called his four scales nominal, ordinal, interval, and ratio, so the system is often called "NOIR." Each category is distinguished by the range of possible values, and the types of comparisons that can be made between values. In GIS and spatial analysis, the scale of a variable determines the types of analytical methods that may properly be employed; in cartography, different types of symbology and different types of Thematic map are used for each scale.

## Nominal Level

Nominal data distinguishes between types or class of data, but they do not have numbers associated with them unless the numbers are used as a numerical identification. Nominal data are observations that have been placed in sets of mutually exclusive and collectively exhaustive categories. Some examples would be tree types or hair color. For tree types, you could have aspen, oak, and pine. For hair color, you could have red, blonde, and brunette. These data sets contain names or color but have no particular order or hierarchy. An example of using numerical identification for nominal data would be zip codes or phone numbers, because they lack a set order or hierarchy. The only comparison that can be made between nominal data is whether they are the same or different.

## Ordinal Scale

Ordinal data are categorical data that have a natural ranking or order. For example, temperature can be classified as "hot", "warm", "lukewarm", "chilly", or "cold," with an inherent order. Another example is the ordinal numbers: 1st, 2nd, 3rd, 4th, etc. Thus, comparisons of "more" or "less" can be made between ordinal data, in addition to the "same" or "different" of nominal data. However, comparisons of the degree of difference cannot be made.

## Interval Scale

Interval data is quantitative or numerical data that is measured on a physical scale without a true origin; that is, the value 0 does not represent the absence of the property. One example is temperature measured in degrees Fahrenheit, because 0 degrees does not mean the absence of any temperature. The degree of difference between two values can be measured through subtraction, thus measuring the interval between the values. For example, if the temperature one day is 20°F and the next day is 40°F, it is meaningful to say that it is 20°F warmer. A ratio between these two numbers is not meaningful; that is, it would not be appropriate to say that the second day is twice as warm as the first day. Other examples of interval measurements are dates and times.

## Ratio Scale

Ratio data is quantitative data in which ratios between two values have definite meaning (eg, 2 is half of 4 and twice as many as 1), and unlike Interval data will have a meaningful zero. The normal numbers (0, 1, 2, 3, etc) is a common ratio data set. A common geographic example of ratio data is density (i.e. population, ethnicity, etc.). Any percent value from 0 to 100 will have a meaningful zero. Another example is temperature measured in degrees Kelvin. Unlike Fahrenheit, 0 degrees on the Kelvin scale actually means the complete absence of thermal (kinetic) energy, and 100 degrees Kelvin has twice as much thermal (kinetic) energy as 50 degrees Kelvin. Meaning that unlike in Fahrenheit, 100 degrees Kelvin is twice as hot as 50 degrees Kelvin.

## Criticisms and Extensions

While Stevens' system of scales has gained wide usage in the scientific community, including GIScience, many have found it inadequate.  The most common criticism is that it is incomplete; that there are many types of data that don't easily fit into the four categories. In order to adjust to this, Nicholas R. Chrisman expanded on Stephen's original typology measurement scale. Chrisman's scale expounds on Stephen's existing categories while suggesting new ones as well. Chrisman's Typology  scale includes:

1. Nominal
3. Ordinal
4. Interval
5. Log-Interval
6. Extensive Ratio
7. Cyclical Ratio
8. Derived Ratio
9. Counts
10. Absolute

### Cyclic Scales

Cyclic number scales are scales that are "bounded within a range and repeat in some cyclic manner." Time on a clock or 360 degrees in a circle are both examples of a cyclic scale. In a 360 degree circle 359 is as close to 0 as 1 is. The listing of seasons is another example where the relationships are invariable, regardless of the starting point (i.e. they can be ordered summer-fall-winter-spring-summer or winter-spring-summer-fall-winter). It is important that cyclic scales be treated differently than normal interval data.

### Hierarchical Classes

A hierarchical classification system is a system of categorizing in which things are categorized according to their rank or position in relation to the other things with which they are classified.    In this type of a scale, the category names are usually nominal but categories can be sub-sets or super-sets of other categories. An example of this type of classification is the Linnaean Classification system in biology where organisms are classified by Kingdom, Phylum, Class, Order, Genus and Species. Another example is the Omernik ecoregion system where a hierarchical scale is used to categorize different ecosystem types. Given Stevens' focus on comparability as the rubric for each scale, hierarchical categories are more than nominal, because one can not only tell whether two individuals are the same or different, but can also measure how similar or different they are (by which higher-order categories they are both in), which makes them almost ordinal.

### Indices

An index is an artificial measure of a property that cannot feasibly be measured directly. It is often constructed by mathematical models based on measurable variables, such as the Dow Jones Industrial Average or the Human Development Index.. It can also be developed by convention (e.g., scoring in sporting events), or even by personal judgement ("rate this hotel on a scale of 1 to 5").

An index is typically unit-less, and is therefore often given in "points." Indices serve to model complex phenomena by combining multiple factors . One example of an index is the well-known Human Development Index (HDI) created by the United Nations Development Programme. It is a measure of achievement, by country, in key fields of human development such as: gross national income, life expectancy at birth, and mean years of schooling received by an individual .

The values of an index appear to be quantitative, and mathematical operations could be performed on them, but they do not have any empirical meaning beyond relative comparison. For example, an HDI score of 0.920 (the score of the United States in 2015) does not mean anything unless one compares it to other countries (it is #10 in the world, and thus considered "highly developed"). Therefore, it is more powerful than ordinal but less powerful than ratio. This distinction means that index variables should be treated as a special case in geographic information techniques such as GIS analysis and thematic maps.

### Other Issues

Stevens' scale of measurement only applies when data within data sets are mutually exclusive. For example, in a nominal data set with black and white as the only options, a feature with 10% gray would either include both options or not fall in the nominal data set.