A practical, topical problem: Consider a typical roll of toilet paper (TP) with perforated sheets of fairly uniform size, and suppose we're interested in the distribution of a sheet's weight $W$ but our scale's precision is markedly larger than $W$'s mean. For example, suppose our (digital) scale reports in increments of 2 g (grams) and $W$'s mean is near 0.34 g. What are efficient and realistic data-collection and -analysis strategies for estimating $W$'s mean and variance? I'm most interested in strategies that are either (a) easy enough for most laypersons -- with little or no statistics experience -- to comprehend and carry out or (b) rigorous in design and analysis (e.g., optimal in a useful sense). I'd appreciate point estimates but especially interval or distribution estimates, and frequentist, Bayesian, or other approaches are fine. Three non-essential remarks:
R1. Perhaps my favorite strategy I've considered is to obtain at least two multi-sheet samples (e.g., 10 sheets weigh 4 g, 20 weigh 6 g) and apply a maximum-likelihood estimator that may resemble techniques for coarsened data. I'm still unsure how best to choose the number and size of samples and obtain them readily (e.g., minimal unraveling of TP roll).
R2. This is inspired by stockpiling of supplies during the COVID-19 pandemic. For instance, to be rational we may wish to estimate $W$'s mean and variance as part of a larger study to predict how long our household's TP will last, perhaps with an expression of error or uncertainty. That's just one of many "everyday" stats projects motivated by the pandemic's novel circumstances.
R3. In R2's study of TP depletion time, we might ignore $W$ and work directly with the weight of full and partial rolls (less any non-TP tube). For instance, simply weighing a household's TP supply before and after a given time period (e.g., a few days) would suffice for a decent point estimate of when that weight will be 0.