Hi Jaydeep,

This is an interesting measurement and analytic problem! Here is an approach which you may find useful.

Imagine creating a series of concentric rings--like an archery target--within the dish. For example, you could create a series of equally-spaced rings: the first (ring #1) having a 4.7mm radius from the center of the membrane; the second (ring #2) having a 9.4mm radius; and so forth (rings could be imposed using a compass on a transparent cover for the dish). Note that these rings are analogous to a (circumferential) Likert-type ordinal score.

If you are interested in finding distance from the center of the membrane in which most bacterial colonies are located, this is a simple matter. The "class" (dependent) variable is number of colonies, and the "attribute" (independent variable") is ring number. A modern machine-learning paradigm--which maximizes classification accuracy, provides an exact p value, estimates reproducibility of the finding, norms the result against chance, and requires no distributional assumptions--may be used to identify the distance which identifies the region (i.e., distance from center) which maximizes the number of colonies.

The number of rings--that is, the precision of your discriminant cutpoint--depends on the size of the colonies, and your ability to accurately assess the number of colonies within each ring. Some colonies may exist within two or more rings, and would be counted as a member of each ring within which the colony was located.

Another variation of the analysis would entail counting the number of colonies which are closer to the center, versus further from the center, for each ring.

Yet another variation would entail simply observing the radius which clearly (via visual examination--a photograph would be instrumental for inspection, and for the publication) demarcates populated versus unpopulated distances (or, even, regions, not necessarily rings), and using the paradigm to discriminate the radius [or region(s)] on the basis of number of colonies.

Here is an open-source paper which demonstrates this methodology:

https://odajournal.com/2016/09/19/novometric-analysis-with-ordered-class-variables-the-optimal-alternative-to-linear-regression-analysis

And, here are brief reviews of the paradigm:

Early Research: https://odajournal.com/2013/09/19/optimal-data-analysis-a-general-statistical-analysis-paradigm

https://odajournal.com/2014/04/06/a-statistical-guide-for-the-ethically-perplexed-chapter-4-panter-sterba-handbook-of-ethics-in-quantitative-methodology-routledge-2011-clarifying-disorientation-regarding

Recent Research: https://odajournal.com/2017/04/18/what-is-optimal-data-analysis

The first issue is that the further you are from the center the greater area you have. So I would expect more colonies further from the center because that gives me the greatest area. So the null-hypothesis is that area explains colony abundance. The alternative hypothesis is that colonies are more abundant closer to the center. Given a source that delivers 2 cfus per square centimeter does the distribution of colonies conform to this application or is the distribution skewed towards greater than expected abundance in the middle?

In ecology the equivalent would be a form of transect sampling called a trapping web: https://pubs.er.usgs.gov/publication/70029575

The issue is that closer to the center the trap density increases up to the point where a trap in the center has a 100% probability of capturing the target that is at the center. Modeling the decline in captures with distance from the center allows estimation of population density.