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# Failure Probabilities from Quality Control Charts

This Demonstration aids in the visualization of how the probability of exceeding a selected tolerance level can be estimated from an irregular time series of the kind frequently encountered in industrial quality control charts. The estimation method assumes that the records consist of independent random entries that are normally or lognormally distributed. The method also translates the series's mean and standard deviation into frequencies of events that are outside the permitted range.

### DETAILS

Snapshot 1: normally distributed record with zero mean and unit standard deviation
Snapshot 2: lognormally distributed record with low logarithmic mean and unit logarithmic standard deviation
Snapshot 2: lognormally distributed record with unit logarithmic mean and low logarithmic standard deviation
The random fluctuations in quality control charts can be translated into a set of probabilities and frequencies of future deviations that are considered unacceptable, provided that the entries are independent, have no trend or long range periodicity, and can be described by a parametric distribution function. If this is the case, the distribution's parameters can be used to calculate these probabilities and frequencies even if deviations exceeding the specified magnitude have not yet been observed. The parametric distribution is commonly normal or lognormal but it may also be of a different type.
This Demonstration generates two sequences of random values, one having normal and the other lognormal distribution. It then calculates the probability of finding values that exceed specified upper and lower limits. The user first chooses the type of distribution and whether to work with one sequence or to generate a new one whenever a control is changed. Subsequently the user specifies the number of entries in the sequence, , the distribution's mean, , or its logarithmic mean, , the corresponding standard deviation, , or its logarithmic standard deviation, , the upper and lower limits ("thresholds"), and , respectively, and the plot's maximum and minimum axis values. The generated sequence is then displayed on the plot as a set of dots connected by a thin colored line with the upper and lower limits shown as two horizontal lines. The theoretical probabilities (in percentages), the theoretically expected number, and the actual counted number in the displayed sequence (of length ) are displayed above the plot.
The number of points in the generated sequence, the distributions' parameters, the upper and lower limits and the display's control parameters are all entered with sliders.
References:
M. Peleg and J. Horowitz, "On Estimating the Probability of Aperiodic Outbursts of Microbial Populations from Their Fluctuating Counts," Bulletin of Mathematical Biology, 62, 2000 pp. 17–35.
M. Peleg, A. Nussinovitch, and J. Horowitz, "Interpretation and Extraction Useful Information from Irregular Fluctuating Industrial Microbial Counts," Journal of Food Science, 65, 2000 pp. 740–747.
M. G. Corradini, M. D. Normand, A. Nussinovitch, J. Horowitz, and M. Peleg, "Estimating the Frequency of High Microbial Counts in Commercial Food Products Using Various Distribution Functions," Journal of Food Protection, 64, 2001 pp. 674–681.
M. Peleg, Advanced Quantitative Microbiology for Food and Biosystems: Models for Predicting Growth and Inactivation, Boca Raton, FL: CRC Press, 2006.

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