Estimating the Time between Mishaps from Quality Control Data

The random entries in the quality control charts of a stable process usually have a normal distribution. This implies that there is a probability that an entry or count exceeds or falls below some threshold level (an event we call a mishap). The mishap's frequency is determined by the normal distribution's parameters and the thresholds. This Demonstration simulates such charts, records the occurrences of mishaps, determines the time intervals between them, and plots their histogram. These statistics can serve as a tool in risk assessment. The histograms of the times to either exceed the upper threshold or fall below the lower one are special cases that can also be examined in the Demonstration.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


Snapshot 1: a simulated quality control chart of a safe process
Snapshot 2: a simulated quality control chart of an unsafe process
Snapshot 3: a simulated quality control chart with a high threshold only (one-sided)
Snapshot 4: a simulated quality control chart with a low threshold only (one-sided)
Industrial quality control (QC) or quality assurance (QA) records of chemical and physical properties frequently resemble a randomly fluctuating time series whose entries have a normal distribution. The fluctuation pattern can be translated into the probability of a mishap, that is, surpassing a given upper threshold or falling below a lower one, using the distribution's parameters, be it normal or some other parametric distribution function [1, 2]. These parameters can also be used to estimate the distribution of the time intervals between mishaps, which can serve as an additional intuitive measure of the process's stability or safety.
In this Demonstration, you can enter the normal distribution parameters and the record's length with sliders. A record is then generated and the mishaps, of either or both kinds, are recorded. (The choices are: upper threshold only, lower threshold only, or both.) Since we assume that the data is entered at fixed time intervals, the mishap's index, i, is also a measure of its occurrence time in the corresponding units. The times between successive mishaps are calculated by the program and their histogram is plotted. When the record is sufficiently long and there is only a single threshold, high or low, the distribution of the times between successive mishaps is expected to approach a geometric or exponential distribution, with a mean determined by the entries' distribution parameters and the threshold level. In the case of two thresholds, the distribution's shape depends on the relative magnitudes of the upper and lower thresholds as well as on the distribution parameters.
For visualization, the Demonstration lets you choose and plot a section of the generated random record by using sliders to choose the indices of the initial and final entries. Superimposed on the plot are the upper and/or lower thresholds drawn as horizontal colored lines. The histogram of the times between successive mishaps is shown below, accompanied by the numerical values of their mean, standard deviation, median, and skewness.
[1] M. Peleg and J. Horowitz, "On Estimating the Probability of Aperiodic Outbursts of Microbial Populations from Their Fluctuating Counts," Bulletin of Mathematical Biology, 62(1), 2000 pp. 17–35. link.springer.com/article/10.1006%2 Fbulm .1999.0112.
[2] M. Peleg, A. Nussinovitch, and J. Horowitz, "Interpretation and Extraction of Useful information from Irregular Fluctuating Industrial Microbial Counts," Journal of Food Science, 65(5), 2000 pp. 740-747. onlinelibrary.wiley.com/doi/10.1111/j.1365-2621.2000.tb13580.x/abstract.
[3] C. Gonzalez-Martinez, M. G. Corradini, and M. Peleg, "Probabilistic Models of Food Microbial Safety and Nutritional Quality," Journal of Food Engineering, 56(23), 2003 pp. 135–142. www.sciencedirect.com/science/article/pii/S026087740200242X.
[4] M. Peleg, M. D. Normand, and M. G. Corradini, "A Study of the Randomly Fluctuating Microbial Counts in Foods and Water Using the Expanded Fermi Solution as a Model," Journal of Food Science 77, 2012 pp. R63–R71. onlinelibrary.wiley.com/doi/10.1111/j.1750-3841.2011.02469.x/abstract.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+