Teaching:TUW - UE InfoVis WS 2008/09 - Gruppe 05 - Aufgabe 1 - Level (Scales) of Measurement

LEVELS OF MEASUREMENT

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DEFINITION

[...] measurement, in the broadest sense, is defined as the assignment of numerals to objects or events according to rules.

[Stevens, 1946]

Measurement of some attribute of a set of things is the process of assigning numbers or other symbols to the things in such a way that relationships of the numbers or symbols reflect relationships of the attributes of the things being measured. A particular way of assigning numbers or symbols to measure something is called a scale of measurement.

[Sarle, 1995]

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INTRODUCTION

Different levels of measurement refer to different relationships among the values that are assigned to the attributes for a variable. Knowing the level of measurement helps when interpreting the data that variable contains.

There are four levels of measurement defined: nominal, ordinal, interval and ratio. These levels are ordered increasingly, different levels of measurements use different mathematical structures, and on higher levels more operations and relations are defined. Each higher level of measurement includes the measurement principle of the lower level of measurement.

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LEVELS OF MEASUREMENT

NOMINAL

At the nominal level of measurement, names are assigned to observations. In most cases numbers are used as names, but also other symbols are possible names. The purpose of this naming is to label or classify the observations.

Two things are assigned the same name if they have the same value of the attribute. Two variable values can only be compared to each other to define if they are equal or not, there are no other relations among the categories such as "less than" or "greater than", and there are no operations such as addition or subtraction.

It is important to mention that these attributes do not imply any ordering among the observations. For example, when numbers are used to represent the different categories, this does not imply anything about the quantitative difference between the categories.

Examples for variables measured on a nominal scale are gender, religion, favorite color, etc.

ORDINAL

At the ordinal level of measurement, all observations are classified in several ordered categories and numbers are assigned to these categories. The categories are orderes from low to high, which is encoded in the numbers assigned to each category. These numbers are called "ordinals". However, the difference betweeen tho categories is not defined at this scale of measurement.

Comparisons like "less than" and "greater than" can be made, and also equality and inequality are defined. However, operations such as addition and subtraction are undefined.

An examples for variables measured on an ordinal scale are academic grades (1, 2, 3, 4, 5).

INTERVAL

Variables measured at the interval level also encode the distance between attributes. All distances between two neighbouring attributes are equal. This means for example that it is possible - and makes sense - to calculate the average of an interval variable, in contrast to ordinal variables where average numbers can not be used. However, ratios can not be applied to interval scales, for example when measuring temparature 30 degrees is not twice as warm as 15 degrees.

As in ordinal scales, relations such as less, greater or equal are defined. Operations such as multiplication and division cannot be calculated, but ratios of differences can be expressed.

"If it's twice as cold today as it was yesterday," runs a popular joke, "and it was zero degrees yesterday, how cold is it today?" This illustrates the limitation of interval measurements such as Celsius and Fahrenheit temperature: by setting zero at an arbitrary point, they make it impossible to multiply and divide meaningfully.

[Wikipedia, 2008]

Examples for variables measured on an interval scale are temperature measured in degrees Celsius or Fahrenheit, or calendar years.

RATIO

The main difference to the interval scale is that at the ratio level of measurement, the ratio between two measurements is meaningful. Therefore there must be an absolute zero value. In that case, operations such as multiplication and division can be carried out as well.

Most physical measures can be used as examples for variables measured on an interval scale, such as weight or height. Other examples are age or amounts of money.

COMPARISON

The four levels of measurement are defined in a hierarchical way. Each level of measurement provides certain qualities, and in addition every level includes all qualities provided by the level below.

[Trochim, 2006]

EXAMPLES

The following table shows some measurements and their level of measurement:

[Wharrad, 2004]

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REFERENCES

[Stevens, 1946] Stanley Smith Stevens: On the Theory of Scales of Measurement. Science, New Series, Vol. 103, No. 2684. (Jun. 7, 1946), pp. 677-680.

[Sarle, 1995] Warren Sarle. Measurement theory: Frequently asked questions. In the Disseminations of the International Statistical Applications Institute, volume 1, edition 4, 1995, Wichita: ACG Press, pp. 61-66.

[Becker, 1997-1999] Lee A. Becker, 1997-1999. Scales of Measeurement. University of Colorado. Retrieved at: 30.10.2008. http://web.uccs.edu/lbecker/SPSS/scalemeas.htm

[Lane, 2003] David Lane. Levels of Measurement. Connexions, June 20, 2003. Retrieved at: 30.10.2008. http://cnx.org/content/m10809/2.4/

[Wharrad, 2004] Heather Wharrad. Levels of Measurement. University of Cambridge. Retrieved at: 30.10.2008. http://www.ucel.ac.uk/showroom/levels_of_measurement/downloads/levels_notes.pdf

[Trochim, 2006] William M. K. Trochim. Levels of Measurement. Research Methods Knowledge Base. Retrieved at: 30.10.2008. http://www.socialresearchmethods.net/kb/measlevl.php

[Wikipedia, 2008] Wikipedia.org, Level of measurement. Retrieved at: 30.10.2008. http://en.wikipedia.org/wiki/Level_of_measurement