arithmetic skills among a population P of pupils: V1: Can pupil (p) perform addition of numbers? No=1; Yes, but only of two-digit numbers=2; Yes=3. V2: Does pupil (p) know the (1-10) multiplication table? No=1; Yes=2. V3: Can pupil (p) perform multiplication of numbers? No=1; Yes, but only of two-digit numbers=2; Yes=3. V4: Can pupil (p) perform long division? No=1; Yes=2. Data collected for the above four variables among a population of school children may be hypothesized to exhibit the Guttman scale shown below in Table 2: Table 2. Data of the four ordinal arithmetic skill variables are hypothesized to form a Guttman scale The set profiles hypothesized to occur (shaded part in Table 2) illustrates the defining feature of the Guttman scale, namely, that any pair of profiles are comparable. Here too, if the hypothesis is confirmed, a single scale-score reproduces a subject's responses in all the variables observed. Any ordered set of numbers could serve as scale. In this illustration we chose the sum of profile-scores. According to facet theory, only in data that conform to a Guttman scale such a summation may be justified. Reproducibility In practice, perfect ("deterministic") Guttman scales are rare, but approximate ones have been found in specific populations with respect to attributes such as religious practices, narrowly defined domains of knowledge, specific skills, and ownership of household appliances. When data do not conform to a Guttman scale, they may either represent a Guttman scale with noise (and treated stochastically), or they may have a more complex structure requiring multiple scaling for identifying the scales intrinsic to them. The extent to which a data set conforms to a Guttman scale can be estimated from the coefficient of reproducibility of which there are a few versions, depending on statistical assumptions and limitations. Guttman's original definition of the reproducibility coefficient, CR is simply 1 minus the ratio of the number of errors to the number of entries in the data set. And, to ensure that there is a range of responses (not the case if all respondents only endorsed one item) the coefficient of scalability is used. In Guttman scaling is found the beginnings of item response theory which, in contrast to classical test theory, acknowledges that items in questionnaires do not all have the same level of difficulty. Non-deterministic (i.e., stochastic) models have been developed such as the Mokken scale and the Rasch model. Guttman scale