# Scales of Measurement

## 1. Overview

Scales of measurement are commonly broken down into four types:
(1) nominal (or categorical),
(2)
ordinal,
(3)
interval, and
(4)
ratio.

The purpose of this set of notes is to briefly summarize several aspects of scales of measurement including:
(a) the measurement principle involved for each scale,
(b) examples of the measurement scales,
(c) permissible arithmetic operations for each scale, and
(d) examples of statistics that are appropriate for each scale.

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## 2. The Measurement Principles

 Nominal Ordinal Interval Ratio People or objects with the same scale value are the same on some attribute. The values of the scale have no 'numeric' meaning in the way that you usually think about numbers. People or objects with a higher scale value have more of some attribute. The intervals between adjacent scale values are indeterminate. Scale assignment is by the property of "greater than," "equal to," or "less than." Intervals between adjacent scale values are equal with respect the the attribute being measured. E.g., the difference between 8 and 9 is the same as the difference between 76 and 77. There is a rationale zero point for the scale. Ratios are equivalent, e.g., the ratio of 2 to 1 is the same as the ratio of 8 to 4.

Each "higher" level of measurement includes the measurement principle of the "lower" level of measurement. For example, the numbers 8 and 9 in an interval scale indicate that the object assigned a 9 has more of the attribute being measured than does the object assigned an 8 (ordinal property) and that all persons assigned a 9 have equivalent amounts of the attribute being measured (nominal property). This also implies that you can do lower level statistics on higher level measurement scales.

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## 3. Examples of the Measurement Scales

 Nominal Ordinal Interval Ratio Gender.Ethnicity. Marital Status. Movie ratings (0, 1 or 2 thumbs up).SES. U.S.D.A. quality of beef ratings (good, choice, prime). The rank order of anything. Degrees F.Most personality measures. WAIS intelligence score. Degrees K.Annual income in dollars. Lengh or distance in centimeters, inches, miles, etc.

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## 4. Permissible Arithmetic Operations

 Nominal Ordinal Interval Ratio Counting Greater than or less than operations. Addition and subtraction of scale values. Multiplication and division of scale values.

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## 5. Examples of Appropriate Statistics

 Nominal Ordinal Interval Ratio [ Crosstabs ] chi square,  c ² phi, f Cramér's V contingency coefficient, CC :lambda, l uncertainty coefficient, UC kappa, k likelihood ratio, LR Goodman & Kruskal tau, t [ Nonparametric ] chi-square,  c ² runs binomial McNemar Cochran Q [ Frequencies ] median, Mdn interquartile range, IRQ [ Crosstabs ] Spearman's Rank Order Correlation, rs (formerly rho) Kendal's Tau-b, tb Kendall's Tau-c, tc Somers' D Gamma, g Mantel-Haenszel [ Nonparametric ] Kolmogorov-Smirnov Sign Wilcoxen Kendall coefficient of concordance, W Friedman two-way anova Mann-Whitney U Wald-Wolfowitz Kruskal-Wallis mean, Mstandard deviation, SD Pearson's product-moment   correlation, r t test, t analysis of variance, ANOVA multivariate analysis of variance, MANOVA factor analysis regression multiple correlation, R coefficient of variation, CFVAR(CFVAR = SD / M)

Notes. The headings within the brackets, e.g., [Crosstabs], identifies the SPSS procedure that that can compute the statistic. The name of the statistic is followed by its symbol, where appropriate. The symbols are from The Publication Manual of the American Psychological Association (American Psychological Association, 1994), pp. 115-118.

You are probably familiar with more of the interval level statistics than nominal, ordinal or ratio level statistics. Most psychological measurement strives to attain interval measurement properties. Consequently we are more likely to teach the statistics that are based on interval level scales of measurement. Ratio level measurement is rare in psychology and sociology.

### References

American Psychological Association. (1994). Publication manual of the American Psychological Association (4th ed.). Washington, DC: Author

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©Lee A. Becker, 1997-1999 -revised 08/26/99