REVIEW ARTICLE On Estimating the Time to the First and Between Successive Accidental Mishaps from Quality Control Records Micha Peleg • Mark D. Normand Received: 9 March 2013 / Accepted: 6 May 2013 / Published online: 9 June 2013 Ó Springer Science+Business Media New York 2013 Abstract In quality control, QC, an accidental mishap can be defined as an entry either falling below a lower or above upper permitted limits without an apparent cause or early indication that this is about to happen. When the randomly fluctuating entries of a QC chart have indepen- dent entries with no trend or periodicity, their distribution can be used to estimate the probability of such a mishap. The entries’ distribution can also be used to estimate the distribution of the times to the first mishap and that of the times between successive mishaps. This is demonstrated with Monte Carlo simulated QC records whose entries’ distribution is normal, representing typical chemical or physical attributes and certain microbial counts, or log- normal, which is frequently encountered in microbial count records. The distribution of the times to cross a lower or upper threshold only is approximately exponential, which has no peak, but whose mean or median can still serve as an intuitive quantitative measure of a process’s degree of risk. When the mishap is defined as crossing either the lower or upper threshold, the distribution of the times between successive mishaps is frequently also exponential but not always. It depends on the entries’ distribution parameters and the thresholds’ magnitudes. In such a case, the most frequent time interval can be identified from the peak of the time intervals’ histogram. The procedures to simulate QC charts having normally or lognormally dis- tributed entries, set the thresholds, plot the histogram of the times to the first mishap or time intervals between suc- cessive mishaps, and to calculate the corresponding exponential distribution have been automated. They have been made available on the Internet as a series of freely downloadable interactive Wolfram Demonstrations. These demonstrations, where all the parameters can be entered and modified with sliders on the screen, can serve as an auxiliary tool to quantify the risk of accidental mishaps in actual industrial food and nonfood processes. They can also be used to simulate hypothetical line performance scenar- ios and examine their consequences. Keywords Quality assurance  Predictive microbiology  Risk assessment  Monte Carlo simulations  Statistical analysis  Random time series Introduction Quality control (QC), or quality assurance (QA), is a useful tool to assure a production line’s satisfactory performance, and a safe product meeting its prescribed specifications. In the food and pharmaceutical industries, recording QC data is not only a common practice [7, 16], but is sometimes required by law. A ‘QC Chart’ is a time record of the final product analysis results or the results of tests performed at selected points during its manufacturing (HACCP). The chart serves as a record of the plant’s compliance with regulations and that the product’s attributes, be they chemical, physical, sensory, and/or microbial, are up to standard and within its specifications. A food product’s specification usually includes a lower allowed limit, for example, % protein, cocoa butter, or fruit contents, an upper limit, for example, % water, fat, connective tissue, or microbial count, or both lower and upper limits, for example, % salt, acid, or sugar. Apart from monitoring compliance with regulations, the QC chart is also used to M. Peleg (&)  M. D. Normand Department of Food Science, University of Massachusetts, Amherst, MA 01035, USA e-mail: micha.peleg@foodsci.umass.edu 123 Food Eng Rev (2013) 5:123–138 DOI 10.1007/s12393-013-9068-1