A measure is a countably additive set function that can take values in a non-negative set or infinity. Its generalizations (which include spectral measures and projection-valued measures) are useful in physics and functional analysis.

Measures are also used in music to denote specific units of musical duration. See the article on musical scales for more information.

Nominal Scales

Nominal scales are used for categorizing variables into labels. These labels don’t have any order, hierarchy, or convey a value. For example, if you were to ask your participants to rate their happiness, satisfaction or level of pain, you would use a nominal scale.

This scale also doesn’t have any arbitrary zero values (like the difference between Fahrenheit and Celsius temperatures), but rather allows you to compare data points that are equidistant from one another. You can perform statistical analyses like mean, median or mode on data recorded on this scale.

An ordinal scale also categorizes variables into categories, but in addition to its labeling properties, it also conveys the order of these values. This makes it easy to analyze data recorded on an ordinal scale using techniques like mean, median or mode. A sub-type of this scale includes two categories only such as a male/female category which is known as a dichotomous scale. This type of scale also doesn’t have any arbitrary or false zero values.

Categorical Scales

The level of a variable’s measurement scale dictates the statistical test type that should be used for it. There are four levels of measurement scales: nominal, ordinal, interval and ratio.

Nominal scales are qualitative and have no numerical value. They are associated with a list of categories that can be labeled, such as “country of birth” or “hair colour”. Nominal data has the advantage of being easy to collect and analyze but does not allow for calculations.

Ordinal scales name groups in a meaningful order (hot to cold, light to heavy, high to low). A Likert scale is an example of ordinal data. Interval scales are also able to record numerical values but they have the ability to calculate a difference between values, such as the differences between 30 and 10 Celsius or credit and SAT scores. In addition, interval scales have a true zero measurement that represents a lack of the characteristic, such as the absence of heat or the absence of weight.

Hierarchical Scales

Artists have long used scale to communicate hierarchical messages in their artwork. For instance, ancient Egyptian and medieval paintings portrayed pharaohs or gods at an exaggerated size relative to mortals in their compositions. Artists use this technique to convey spiritual or political power and authority, which is often symbolized by the relative height of a figure within the painting.

The most basic measure of data is the nominal scale, which consists of categorical or ordinal data. Interval scale is a subset of this scale and contains numeric values that can be ranked based on frequency of observations. Mode and median can be computed for interval data too.

In most cases, ecological patterns and processes have characteristic scales that are intrinsic to the phenomena of interest (Bloschl and Sivapalan 1995). These characteristic scales may be related in space or time or characterized as organizational or integrative levels within a hierarchy of entities or events. However, it is important to note that detectable characteristic scales are often tinted with observer subjectivity.

Descriptive Scales

Scales can be used in qualitative observational data to describe the qualities of points on a point set (called a vector). There are four scale types: nominal, ordinal, interval and ratio. Each has properties that determine how the scale should be analysed. These include identity, magnitude, equal intervals and a true zero. The temperature scale is an example of an interval scale.

Questionnaires with rating scales, Likert scales and other ranking questions are ordinal scales. They also have an order and the responses can be compared with each other.

Internal consistency of items is often used to measure unobservable concepts. A popular way to do this is by calculating Cronbach’s alpha, which measures how well the different responses are correlated with each other. This is particularly important when using ratings scales where the results can cluster around 1 or 5. The use of open-ended questions e.g. a comment box, can avoid this but requires careful analysis to extract meaningful insights.