# Measurement level of percentile scores

This one is bothering me for a while, and a great dispute was held around it. In psychology (as well as in other social sciences), we deal with different ways of dealing with numbers :-) i.e. the levels of measurement. It's also common practice in psychology to standardize some questionnaire, hence transform the data into percentile scores (in order to assess a respondent's position within the representative sample).

Long story short, if you have a variable that holds the data expressed in percentile scores, how should you treat it? As an ordinal, interval, or even ratio variable?!

It's not ratio, cause there no real 0 (0th percentile doesn't imply absence of measured property, but the variable's smallest value). I advocate the view that percentile scores are ordinal, since P70 - P50 is not equal to P50 - P30, while the other side says it's interval.

Please gentlemen, cut the cord. Ordinal or interval?

• By the way, P70 - P50 represents the percentage of people who are between the 70th percentile and 50th percentile and that percentage is 20. Clearly that is the same as P50 -P30. When assessing if differences are equal I do not think you should look at the underlying scores. – user28 Jul 28 '10 at 18:44
• And the same stands for calculating correlation coefficients, I guess? – aL3xa Jul 29 '10 at 0:26
• Yes, that would be correct. In fact correlation would be ratio as 0 means no correlation and such a conclusion is scale invariant. By the way, I suspect that percentiles would also be classified as ratio as the 0 point is scale invariant but it does not really matter. Most statistic applications require interval level measurements not necessarily ratio. – user28 Jul 29 '10 at 0:53

The critical property that distinguishes between ordinal and interval scale is whether we can take ratio of differences. While you cannot take ratio of direct measures for either scale the ratio of differences is meaningful for interval but not ordinal (See: http://en.wikipedia.org/wiki/Level_of_measurement#Interval_scale).

Temperature is the classic example for an interval scale. Consider the following:

80 f = 26.67 c

40 f = 4.44 c and

20 f = -6.67 c

Differences between the first and the second is:

40 f and 22.23 c

Difference between the second and the third is:

20 f and 11.11 c

Notice that the ratio is the same irrespective of the scale on which we measure temperature.

A classic example of ordinal data is ranks. If three teams, A, B, and C are ranked 1st, 2nd, and 4th, respectively, then a statement like so does not make sense: "Team A's difference in strength vis-a-vis team B is half of team B's difference in strength relative to team C."

Is ratio of differences in percentiles meaningful? In other words, is the ratio of difference in percentiles invariant to the underlying scale? Consider, for example: (P70-P50) / (P50-P30)?

Suppose that these percentiles are based on an underlying score between 0-100 and we compute the above ratio. Clearly, we would obtain the same ratio of percentile differences under arbitrary linear transformation of the score (e.g., multiply all scores by 10 so that the range is between 0-1000 and compute the percentiles).

John Tukey strongly and cogently argued for a proportion type of measurement in his book on EDA. One thing that makes proportions special and different from the classical "nominal, ordinal, interval, ratio" taxonomy is that frequently they enjoy an obvious symmetry: A proportion can be thought of as the average of a binary (0/1) indicator variable. Because it should not make any meaningful difference to recode the indicator, the data analysis should remain essentially unchanged when you re-express the proportion as its complement. Specifically, recoding $0\to 1$ and $1\to0$ changes the original proportion $p$ to $1-p$. For example, it should make no difference to talk about 60% of people voting "yes" or 40% voting "no" in a referendum; the two numbers 0.6 and 0.4 represent exactly the same thing. Thus, statistics, tests, decisions, summaries, etc., should give the same results (mutatis mutandis) regardless of which form of expression is used.

Accordingly, Tukey used re-expressions of proportions, and analyses based on those re-expressions, that are (almost) invariant under the conversion $p\longleftrightarrow 1-p$. They are of the form $f(p) \pm f(1-p)$ for various functions $f$. (Taking the minus sign is usually best because it continues to distinguish between $p$ and $1-p$: only their signs differ when re-expressed.) When scaled so that the differential change near $p=1/2$ equals $1$, he called these the "folded" values. Among them are the folded logarithm ("flog"), proportional to $\log(p) - \log(1-p)$ = $\log(p/(1-p)$ = $\text{logit}(p)$, and the folded root ("froot"), proportional to $\sqrt{p} - \sqrt{1-p}$.

A mathematical exposition of this topic is less convincing than seeing the statistics in action, so I recommend reading chapter 17 of EDA and studying the examples therein.

In sum, then, I am suggesting that the question itself is too limiting and that one should be open to possibilities that go beyond those suggested by the classical taxonomy of variables.

### Addendum (August 2014): Why "Interval" and "Ratio" are not quite correct answers

Stevens created the nominal-ordinal-interval-ratio typonymy in a cogently argued 1946 paper in Science (New Series, Vol. 103, No. 2684, pp 677-680). The basis for the distinctions is explicitly invariance of the "basic empirical operations" under group actions. His Table 1 describes the relationship between scale and group thus:

\eqalign{ &\text{Scale}&\text{Mathematical Group Structure} \\ &\text{Nominal}&\text{Permutation Group } x^\prime = f(x);\ f(x) \text{ means any one-to-one substitution.} \\ &\text{Ordinal}&\text{Isotonic Group } x^\prime = f(x);\ f(x) \text{ means any monotonic increasing function.} \\ &\text{Interval}&\text{General Linear Group } x^\prime = ax + b \\ &\text{Ratio}&\text{Similarity Group } x^\prime = ax }

(This is a direct quotation, with some columns not shown.)

This must be read with some latitude, because we always have the option of choosing a model that is not exactly correct. (For example, a Normal distribution as a model of variation can be extremely useful and quite accurate even when applied to, say, the heights of people, which can never be negative even though all Normal distributions assign some probability to negative values.) Thus, for instance, data of extremely small proportions could arguably be considered as being of ratio type because the upper limit of $1$ is practically irrelevant. Data of very closely spaced proportions that approach neither of the limits $0$ or $1$ might conceivably be considered of interval type. Limiting the scope of the questions to either of these special cases would (partially) justify some of the other answers in this thread which insist that proportions are on an interval scale or ratio scale. However, when proportions in a dataset can be both large (greater than $1/2$) and small (less than $1/2$) and some of them approach $1$ or $0$, then obviously neither the general linear group nor the similarity group can apply, because they do not preserve the interval $[0,1]$. This is why Stevens' classification is incomplete and why usually it cannot be applied to proportions.

Continuous (interval); this is a method how to convert ordinal data to something that may have some distribution that makes sense.

• Indeed we have suggested something like this in this topic; I'll dig for a link. – user88 Jul 28 '10 at 18:05
• And I think your criterion for 'intervality' is valid only for a uniform distribution. – user88 Jul 28 '10 at 18:10