Sums and Sieves: Estimating Unknown Variance and Correlation Values in the Interpretation of Combinations of Percentile Values

Occupational Applications

This article describes techniques to evaluate combinations of percentile values when only the nominal percentile level and percentile values are known, but not the underlying data. The methods described can be employed to estimate anthropometric accommodation. One application is that of estimating accommodation from the sum of two or more percentile values when a desired dimension has not been directly measured, but can be estimated by combining the percentile values of two or more known percentile values (e.g. the percentile for seated eye height above the floor as the sum of 90th percentile values of seat height and eye height above the seat). A second application is to estimate the joint accommodation on multiple variables when two or more percentile values have been used to define dimensions (e.g. what is the joint accommodation for a chair seat when the height, depth, and width are defined by their respective 90th percentile values?).

Technical Abstract

Background: Several authors have noted the utility of combining percentile values, for example as sums of server response times, or as the proportion of cases in the intersection of sets of percentile-based parameters of objects, such as furniture dimensions. Purpose: This article describes methods to estimate unknown variances and correlations as a function of the percentages or quantiles of a distribution. Methods: Two methods are presented with which unknown variances and correlation values can be estimated. Variances are found by converting percentiles or quantiles to indicator function variables. As a result, the variance can be expressed as a function of the quantile. Similarly, there are limits on the range of correlation values, which are a function of the percentile or quantile. The midpoint of the range of possible correlation values can be used to estimate combinations of percentiles, as it minimizes the possible error due to estimates of the correlation value. Finally, estimates of combinations of percentiles are made using the derived variance and estimated correlation values. These estimates are compared with observed combinations. Results: The maximum possible error using these methods of estimating variances and correlations for combinations of two variables are illustrated for a selected range of quantiles. The maximum possible error is least at the extremes of the quantile range (e.g. for quantiles greater than the 90th or less than the 10th). Conclusions: In some instances where only limited data are available, such as a single pair of percentile values, it may be useful to draw some conclusions regarding the general population. The techniques described in this article allow for estimating the proportion of the population whose measurement values are concurrently less than or equal to the specified percentile values; or, to estimate the proportion of the population whose summed measurements are less than or equal to the sum of the known percentile values.

Keywords:

• Adding percentiles
• accommodation estimates: interpreting combinations of percentiles