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Background

I am developing single use sensors to test a concentration of an analyte in a sample. I am interested in the reproducibility of the individual sensor batches and the subsequent probability to find results of single sensors of this batch within an specific range.

Setup

I am producing a batch of sensors (~2000) and then test 5 sensors. The sample results give me a mean and a sample variance. Now I am interested to calculate the probability of any single sensors measured in the future to give me a result with a deviation larger then 10% form the expected value.

Calculation approach

I estimate the population variance from the n=5 sample and then assuming a normal distribution for any future measurements. I confirmed the normality by a shapiro wilk test of the sample residues.

Next I calculate the Z value of my 10% boundary from the mean and the estimate of the population variance from the sample and derive the corresponding probability. I have particular troubles how to deal with the situation that I want to predict the result of just one sensor, no average of several sensors which would trigger me to use a T statistic given the small sample size.

I also considered to use the Chi Squared upper 95% CI limit for the sample variance as estimate for the worst case population variance for calculation of the Z statistic, which results obviously in a higher probability to find a sensor outside of the 10% boundary.

Questions

  1. Is it correct to assume normal distribution of future single measurement and hence to use a Z statistic?
  2. Is the estimate for a population variance from a n=5 sample suitable for calculating the Z statistic or do i need to correct somehow for the small sample size?
  3. Is a Chi Squared derived upper 95% CI limit for the sample variance more precise in estimating the population variance when i am interested in the worst case scenario?
  4. Is there an alternative approach to the problem

I hope I have described the problem detailed enough and I am really looking forward to your input.

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Welcome to the site!

Without more information, yes, a normal distribution sounds fine. There are scenarios where other models work better, but this is if nothing else a good, standard starting place.

I'd recommend what you mentioned in #3, correcting for population size as you are constructing the Gaussian model; Use the worst mean/stddev for a given % chance occurrence. (Don't forget to pessimistically estimate mean too, using that same fudge factor.) Then compare how bad those "worst 0.x% of items" are expected to be vs requirements.

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  • $\begingroup$ Dear Andrew, thanks for having me and confirming my approach. If I understand you correctly I should derive a fudge factor, like the 95 CI limit from a t statistic on the mean, right? To give some additional information: I use the same sample set to calibrate the sensor batch so the mean of this sample will be equivalent to the requirement. I am not sure how to integrate a pessimist mean. I have thought about using the t test derived 95CI of the sample mean as a measure of spread of the population. I just not sure how to work with this in predicting p of future results. $\endgroup$
    – Fons
    May 13, 2020 at 18:29
  • $\begingroup$ Correct. The overall idea is whatever method you use to "be pessimistic about" your variance should be consistent with how you estimate the mean. $\endgroup$ May 13, 2020 at 19:04

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