Variability, Measure of

Variability, meaning differences, is a critical construct in research and science. Objects and events that are constant do not require prediction or explanation. The advancement of science depends on the extent that the differences between objects and events are explainable and predictable. The goal of the scientist and researcher is to create parsimonious scientific models that predict the variability between objects and events and to test those models against the real world. The ability of the scientist and researcher to measure variability allows the assessment of competing scientific models and theories about the world.

There is a confusing array of symbols in the statistical world, all measuring variability. When referring to a theoretical probability model, the variability is symbolized by VAR(X) and might be described with Greek letters. With sample data, variability is measured conventionally using statistics such as the standard deviation, variance, and range. If the measure of variability is the standard deviation or variance, the variability is generally symbolized by the letter s. Measures of sample variance are used as estimates of model variability. Wide varieties of subscripts are used with both model and sample measures of variability to clarify the meaning of the measure. All measures share certain common elements and interpretations.

Theoretical probability models (probability distributions) are mathematical equations used to model distributions of real-world objects or events. In the case of theoretical probability models, variability has a precise definition. Because model parameters are most often symbolized by Greek letters and the variability of a theoretical probability model can often be expressed as functions of these parameters, theoretical model variability is often expressed as equations with Greek letters.

If a sample of tenured full professors was taken and each was asked about the number of hours per week they worked in their academic position, considerable variability would be found in the data. Some professors might work the absolute minimum required by their respective colleges and universities, whereas others might maintain long academic work weeks. The [Page 1600]variability of the data could be measured either by using conventional statistics or, alternatively, by using the variance of the probability distribution created to model the data. If a normal distribution is used to model the data, then the conventional statistics and model values converge. If a normal curve model was found to be unacceptable because it is not possible to work negative hours in a week, then other measures of variability might be more appropriate.

The first step in the scientific method is to model the distribution and to quantify the amount of variability in the data. The second step is to create models to predict the data. In most cases, at least when the model works, the initial variability will be reduced. In the tenured professors example, the researcher might create a prediction model based on variables such as the professor's department, age, years to retirement, health, and publication record. To the extent that the prediction model works, the differences (variability) between the predicted and the observed data will be small. Many statistical methods rely on a mathematical property of variability where variance measures can be broken down into component parts that are additive and might be summed to the whole. These methods work by “partitioning” the existing variability and then by interpreting each component. In the example, the total variability could be partitioned into that which could be predicted by the model and that which cannot.

...