# Weighing toilet paper with an imprecise scale

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.

• As hinted at in R1, this question may be deceptively simple. For instance, certain sample sizes will yield badly biased estimators, TP sheets may be sampled in numerous ways (e.g., contiguous or not), and sampling more sheets involves tradeoffs. – Adam Hafdahl Mar 28 '20 at 21:32
• 1st, it's important to know (or have a reasonable model for) the behavior of your digital scale. (a) Does it weigh with reasonable accuracy and precision and then round to the nearest 2g? (b) Is it unbiased with a SD of about 1g, reporting to the nearest ± 2g? (c) Does high bias but excellent precision. (c) Etc. // Second, similarly what do you know or assume about W? All sheets very nearly the same weight within a roll, or highly variable? Etc.// Huge differences among various brands here! // I'd weigh an entire roll, count nr of sheets, divide, and move on to something easier to model. – BruceET Mar 29 '20 at 8:07
• "or 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. " I do not see the connection between toilet paper weight and how long it lasts. – Sextus Empiricus Mar 29 '20 at 16:12
• Thanks, @BruceET. I see the potential value of knowing more about the scale's behavior and W's distribution, but in a sense my question's about how to use clever data collection and analysis to learn the latter (about W) and cope with the former (about the scale). Also, surely we can be more efficient than counting an entire roll's sheets. I'm modeling plenty of other things; this question's about TP weight and associated measurement puzzles. :-) – Adam Hafdahl Mar 30 '20 at 13:46
• @Sextus Empiricus: For example, suppose someone concerned about their household's TP supply guesses or estimates that in a typical day about 45 to 50 sheets are used, and they have 2 partial rolls and 4 full rolls of the same TP on hand but don't know how many sheets these rolls contain (e.g., without the packaging). From that usage rate, a TP sheet's mean weight, and the rolls' weight, we could estimate time until TP depletion at least crudely without, say, careful modeling of W's distribution or uncertainty due to sampling or measurement error. – Adam Hafdahl Mar 30 '20 at 14:04