# Summarizing a distribution and calculating CI of mean under measurement uncertainty

I have a process wherein human operators (Ops) evaluate some quantity/metric (for simplicity let's say length of a leaf fallen from a tree) for 1000 representative samples and note down it's value on a form.

Main goal is to estimate the 95% CI of mean length of leaf. I can use the usual formula (mean +- z*(standard deviation of observed length)/sqrt(sample size) but...

The evaluation is subject to measurement errors - measuring instrument not calibrated properly, incorrect value noted down on form, leaf may have withered a little bit, skill difference of Ops in doing the measurement

In order to better quantify the measurement errors, the Ops were divided in to 2 mutually exclusive groups and each of these groups measured the same set of leafs - measurement times might be apart by a couple of hours. Within a group a leaf was measured by only one operator.

So the data looks like following:

sampling_id|length_groupA|length_groupB

Note length_groupA - length_groupB and it's absolute difference is zero for 90% of the sampling_id's. But the remaining 10% varies all over the place. Standard deviation of either of length_groupA or length_groupB is much lower than standard deviation of (length_groupA - length_groupB) which again is much lower than standard deviation of |(length_groupA - length_groupB)|

I'm aware of concept of standard error of measurement (SEm) as provided here

2 questions:

a) I want to graphically summarize the distribution of leaf lengths and intend to use boxplot. Would like to add a caveat that there is measurement error. Should I do boxplot on only one of "length_groupA" or "length_groupB" (and supplement it with standard error of measurement for the difference between groupA and groupB for the same sampling_id). Or for each sampling_id should I randomly select length from one of these 2 groups and do a boxplot on that? Does the later option need to be supplemented with SEm?

b) Now to the primary goal - arrive at 95% CI of mean length. Is there an adjustment to the standard error of the mean formula to include standard error of measurement?

PS: I have tried to simplify the problem; in reality this needs to be independently analyzed for different types of tree - maple, oak, poplar etc

• This is the day-to-day business of basic practical physics courses. One measures some quantity, with some uncertainties introduced by the measurement (e.g. apparatus) itself. One then calculates the final outcome, including all errors. The keyword is "error propagation". What are you most interested in the answer -- reference, explanation or example? – cherub Jun 14 '18 at 12:53
• Thanks for the keyword. math.stackexchange.com/q/754679 has the analytical solution for CI of mean length based on assumptions around the underlying quantity, leaf length - normally distributed + constant variance - neither of which works in my case. Is there a numerical solution available to adjust the CI of mean given measurement uncertainty – Hari Jun 14 '18 at 19:32