# How can I approach this data that has a different total number of results between samples?

I was not sure whether I should post this in Biology Stack Exchange or Math Stack Exchange so I posted it in both.

I was planning on doing a small research for my math class (a final project for the year) on population dynamics. I am going to use yeast cells and I will approximate the population of cells per medium (there will be several containers with yeast cells) using a colorimeter and a microscope. I will then graph the results to find a logarithmic regression for the graph that best represents the data, then I'd find the growth rate of the population of yeast cells. Up to here I have no issues.

I also wanted to calculate a distribution curve for the size of the yeast cells at three different points of the growing phase, t1, t2 & t3 (I would take a sample of the mediums with yeast cells, and using a microscope and a grid I would approximate the size of the yeast cells and separate them by size). The problem is that each sample will most likely have a different total number of cells (also varying depending on the medium). This would mean that the samples I take at, for example, t1 , will have a different total number of cells in them, thus making it impossible for me to average the number of cells with a size between two intervals. I do not want to average the total number of cells mainly because a sample might have 50 cells in it meanwhile another might have 200 cells in it.

Additionally, I wanted to make a binomial distribution too, by classifying the cells as either dead or alive (at t1, t2 & t3). I would calculate the probability of a cell being dead or alive by calculating the ratio between alive cells and total number of cells (I am not sure if this is also correct). Nonetheless, I face the same problem here: there will be different values regarding the total amount of cells per sample.

My question is: should I average the total number of cells per sample? Or is there some other way I can approach this problem?

• I don't understand your paragraph 2. Are you trying to compare the mean size of cells at different times? Then, why can't you just compute the mean (whether the sample are of different size or not). Are you thinking about making some stat test? What test do you have in mind? – Remi.b Jan 9 '19 at 3:18
• Similarly to the second paragraph, I don't understand what is the problem you are phasing. Maybe because I misunderstand what you are trying to do. Having a different sample size will affect your standard deviation for each sample, but you can still compute unbiased mean size and unbiased fraction of dead cells. A priori, the fraction of interest will be number of dead cells over total number of cells (unlike what you suggest), – Remi.b Jan 9 '19 at 3:20
• In all case, this question is not a good fit for Biology.SE (nor is it for math.SE). If I understand you correctly, your problem is a problem of data analysis and the question should rather be asked on stats.SE. I am voting to close as off-topic. – Remi.b Jan 9 '19 at 3:22
• The OP is right that it may not be valid to directly compare correlated samples of unequal size. One way to approach the problem is to use mixed effects models. For the size measurements you could use a linear mixed model, whereas for the probability of survival a generalised linear mixed model with the binomial distribution (logistic regression model). Ideally, you could combine both models into one multivariate model that takes everything into account. – vkehayas Jan 9 '19 at 20:34
• The correct stack for this question is neither Biology nor Math, but rather stats.stackexchange.com. – vkehayas Jan 9 '19 at 20:35