Height of a table: what's the uncertainty? I would like to introduce high school students to error analysis with a simple experiment: find $g\pm \Delta g$ by measuring the time a ball takes to fall from a table, repeating many times.
Uncertainty of time of fall $\Delta t$ is determined statistically (standard error), since the measurements of $t$ must follow a normal distribution.
How about the height $h$ of the table, which is measured with a meter stick? $\Delta h$ is mainly due to the fact that the stick is never perfectly vertical during measurement, so the value read is necessarily greater than the true value. Thus it makes no sense to take $h$ to be the average of the values read, and since the measurements don't follow a normal distribution I don't know how to evaluate $\Delta h$ statistically. How would you find $h \pm\Delta h$?
 A: Uncertainties almost never have a unique cause. The vertical misalignment of the stick can be considered a systematic error, as a parallax error. In principle, it can be reduced below other sources of uncertainty by a careful experimental protocol or changing the way the height is measured. Still, I would expect some residual uncertainty, due to many other reasons (for example a not uniform height of the table or uncertainty in the reading of the length).
However, coming to your question, there is no difficulty in a direct evaluation of the spread around the average value, simply performing enough measurements to have a statistically meaningful sample of different values.
A: In this case the experiment will be a good illustration not only of the uncertainty, but also of the bias of an estimator - one could perform rather symple analysis for small angles. This however may be indeed  abit too much for your students.
A reliable way to introduce uncertainty is to use a chronometer that is started and stopped by humans - since human reaction time is some fraction of a second and noticeably varies from a persont to another. Time of a ball falling from a table is probably too short for this, but a ball falling from a building or rolling in a tilted pipe may be long enough for meaningful data.
A: Let $H$ denote the random variable for the measurement of the height and $h$ a specific value for $H$.  Take a set of $n$ measurements for $H$: $h_1, h_2, ...h_n$.  I suggest using a variety of "rulers" to measure height to compensate for bias inaccuracies in the rulers themselves.  Estimate the mean of $H$ , $\mu_H = {{\sum_{i = 1}^{n}h_i} \over n}  $.  Estimate the standard deviation for $H$ as $\sigma_H  = \sqrt{ {\sum_{i = 1}^{n}{(h_i - \mu_h)^2} \over {n - 1}}}$.  Estimate the standard deviation of the mean of $H$ as $\sigma_{\mu H} = \sqrt{ {\sum_{i = 1}^{n}{(h_i - \mu_h)^2} \over {n(n - 1)}}}$.  The result is $\mu_H \pm \sigma_{\mu H}$ (what you call $h \pm \Delta h$).  Similarly, you collect data for the measured time $T$ and report the result as $\mu _T \pm \sigma_{\mu T}$ where $\mu_T$ is the mean and $\sigma_{\mu T}$ is the standard deviation of the mean.  I suggest using a relatively large height so the relative error in the ruler measurement is lessened, and so the error associated with starting and stopping the timer (especially if manual) is of less importance.
Let $G$ be the random variable for the measured gravity. $G$ is a function of $H$ and $T$: $G = {2H \over T^2}$.  The mean of $G$, $\mu_G = {2\mu_H \over \mu_T^2}$. You estimate the standard deviation of the mean of $G$ using a Taylor series expansion of the function $G = {2H \over T^2}$ about its mean. You can assume $H$ and $T$ are independent to evaluate the result for $\sigma_{\mu G}$, the standard deviation of the mean of $G$.  Your result is   $\mu_G \pm \sigma_{\mu G}$. The details for the series expansion can be found in standard tests for error propagation such as Data Analysis for Scientists and Engineers by Meyer, or online.
A: I think that one should show that the error on the height of the table, between many measurements is considerably smaller than the error on the time of fall, which, of course, should be done first, to have an order of magnitude of $\Delta t/t$
Then one should measure $\Delta h/h$ for many measurements, find that it is much smaller and explain to the students that it is useless and counterproductive to look for details of a much smaller error. Any value of $h$ within the very small spread would do, except perhaps extreme ones, which would be obvious experimental errors, considering the much larger value of $\Delta t/t$.
This, I think, is the proper approach to error analysis. Insisting on details for $\Delta h/h$ would confuse your students. Indeed $\Delta h/h$ is probably smaller than the error on $\Delta t/t$ depending on whether you keep all the experimental values of $t$ or eliminate those too far away from the mean, which are obvious experimental errors.
