Sample Size

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    Rebecca Vinson

    I read an article in Medical Device & Diagnostic Industry Magazine about determining minimum sample size for validations.  They selected a minimum acceptable value and then determined the sample mean and standard deviation of 10 test samples.  They then introduced a variable NSD, the number of standard deviations that the minimum acceptable value is away from the sample mean.  They then took that value to a K-value table to determine the sample size.
    What I don’t understand is where did the K-value table come from.  I can’t find it in any of my statistics books or on the web.  The article mentions that “K values are a convenient way of expressing the statistical interaction of confidence intervals, population percentages, and sample size.”  That also brings up the question of population percentages.  The article defines them as the “proportion of the product exected to exceed the minimum”; however, I have never heard that term before either.
    Could somebody help me out here?
    Thank you!



    Hi Rebecca,
    I believe they are talking about “statistical tolerance limits.”  Assuming the process average (mu) and process standard deviation (sigma) are known, then the middle 99.73 percent of a process’s output will fall between the following limits (assuming a normal distribution):
    mu +  3 sigma
    Unfortunately, we seldom know the true values of mu and sigma and must estimate them with mu-hat and sigma-hat, respectively, based on some sample.  Due this this uncertainly in both process parameters, we would conservatively predict that the middle 99.73 percent of the process’s output is likely to lie between boundaries that are somewhat larger than those given above when mu and sigma are known.  To accomodate this uncertainty, a factor called K (which would be greater than 3 in this case) is used in place of the “3” above.
    mu-hat + K sigma-hat
    In addition to the sample size used to estimate mu and sigma, K depends on the percentage of the process output you wish to bound, as well as the confidence level desired.  For example, when a sample size of 25 is used to estimate mu and sigma, we can say with 95 percent confidence that the middle 99.73 percent of the process output lies within the following interval (for this particular combination, the K factor is 4.02):
    mu-hat + 4.02 sigma-hat
    K factors for other combinations can be found in the references listed at the bottom of this message.
    To address your situation: suppose you have an idea on the value of sigma (from past runs, perhaps) and that you wish to have a given degree of confidence that the process average (which may change from run to run) is at least K sigma above a minimum acceptable value.  Working backwards, you could look up the K value in one of the K tables and find the corresponding sample size, which would be the minimum required to meet the above criteria.
    Hope this helps.
    Wald and Wolfowitz, “Tolerance Limits for a Normal Distribution,” Annals of Mathematical Statistics, Vol. 17, 1946, pp. 208-215
    Owen, D.B., Handbook of Statistical Tables, 1962, Addison-Wesley
    Montgomery, D., Introduction to Statistical Quality Control, 2nd edition, 1991, John Wiley & Sons, pp. 403-405
    Natrella, M., Experimental Statistics: National Bureau of Standards Handbook 91, 1963, U.S. Government Printing Office, pp. 2-13 to 2-15 


    Chip Hewette

    Are you speaking of validation of a device or medicine to be equivalent to a currently used device or medicine?
    For example, let’s say a currently used device can detect 10 cases of cancer when there are truly 12 cases of cancer.  Based on this poor performance someone makes a new device to find the 12 true cases of cancer.  Are you looking for information to calculate the minimum sample size to determine if the new device’s ability to find cancer is at least equal to 10 out of 12?


    Thomas C. Trible

    Rebecca:You can find “K Tables” in Juran’s Quality Control Handbook, 4th Edition – page Appendix II.36 and also on page 691, Table A.7 of Probability and Statistics for Engineers and Scientists, by Walpole and Myers.K Tables are used by reliability engineers to calculate confidence intervals for different reliabilities. Here’s an example: Question; Find the 99 percent tolerance limits that will contain 95 percent of (anything) produced by (some process), assuming an approximate normal distribution. You first calculate mean and standard deviation from n samples of a normal population, then use the K Table to find a k value – from the desired tolerance limit and confidence interval that you want from n samples – and multiple the tabular k value by the sample standard deviation. Add and subtract that product from the sample mean to find, for example, the 99percent tolerance limits about the sample mean.Hope this helps. Walpole and Myers give a fair discussion of tolerance and confidence intervals.

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