Let's say I measure the diameter $(x)$ of five checkers pieces, using a ruler with a precision of $\pm 0.5mm$, and get the following results:
$x_1 = 50 \pm 0.5mm, \; x_2 = 51 \pm 0.5mm, \; x_3 = 48 \pm 0.5mm, \; x_4 = 50 \pm 0.5mm, \; x_5 = 49 \pm 0.5mm$
The average length of the checkers pieces is then $\bar x = 49.6 mm$, with a standard error on the mean of $\sigma_x = 0.51$. I could therefore quote my mean value as $\bar x = 49.6 \pm 0.51 \,[mm]$
How do I incorporate my measurement uncertainties? Should I just propagate them through the calculation of the mean, or is there another approach?
