# Calculation of the probability to find a test result of a single measurement outside a defined range from small sample size

## 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.

• Perhaps a perspective from the SPRT literature, see, for example, repository.upenn.edu/cgi/… . May 13, 2020 at 16:45