Effects of nonnormal distributions on highway construction acceptance pay factor calculation

Moin Uddin, Kamyar C. Mahboub, Paul M. Goodrum

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


Percent within limits (PWL) is a commonly used quality control/quality assurance measure of highway pavement materials and construction, and it is a popular index for adjusting pay factors. However, PWL is based on the assumption of normal distribution of quality characteristics (e.g., concrete compressive strength and asphalt air voids). Skewness and kurtosis, which are common forms of statistical nonnormal distributions, can potentially bias the acceptance pay factor calculations. To examine this potential pay bias, simulations were performed to investigate the magnitude and the direction (overestimation or underestimation) of pay factor calculations. The study revealed that for both one-sided and two-sided specification limits, bias in pay factors not only did vary in magnitude but also reversed in direction over various ranges of PWL. These analyses showed that for a one-sided upper specification limit, on average, a positive skewness and kurtosis can underestimate the pay factor of an acceptable quality level population by 0.90%, and overestimates a rejectable quality level population by 3.8%. This leads to falsely penalizing acceptable products and rewarding bad products. The same was true for two-sided limits, which again varied based upon the percent of defective materials at the tails of the distribution. This is a very important issue because these biases in pay factors can easily upset the relative profit margins of the contractor. Furthermore, this may not be easily detectable without a detailed and sophisticated analysis as outlined in this paper. For multiple quality characteristics based pay factors, analyses showed that the combined magnitude of these biases was not linearly cumulative. Findings of the study indicate that bias in pay was higher for lots with fewer sublots and higher skewness and kurtosis.

Original languageEnglish
Pages (from-to)108-118
Number of pages11
JournalJournal of Construction Engineering and Management
Issue number2
StatePublished - Feb 2011


  • Bias
  • Kurtosis
  • Percent within limits
  • Quality control
  • Skewness

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Building and Construction
  • Industrial relations
  • Strategy and Management


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