I'm currently facing a huge problem in manipulating raw data from an experiment.
Suppose I have the set of data: $$T_1=6.12 \pm 0.001s,\\ T_2=4.90875 \pm 0.001s, \\ T_3=3.9 \pm 0.001s, \\ T_4=3.69 \pm 0.01s$$ Height $h$ is $20m$. Suppose then that the velocity $v$ is $h/t$. How would I go about computing ${v}^{2}$ while taking into account the uncertainties?
The error compounds, but you can still give error bounds. For example in the first case we have $$\frac{20}{6.121}\le v\le\frac{20}{6.119}$$ so $$\left(\frac{20}{6.121}\right)^2\le v^2\le\left(\frac{20}{6.119}\right)^2$$
In decimal form this gives you $$10.676<v^2<10.684$$