round-off error of sample

Question

Basic question:

If I have a large set of numbers where the third decimal has been rounded off. For example

$a_1 = 5055.29$

$a_2 = 1755.13$

$a_3 = 1083.03$

$a_4 = 335.99$

$(...)$

If I sum all observations to one number $S = \sum_{i} a_i$

What is the error term for $S$?

Answer 1

If you want the worst case error, the error in each term is at most $\pm0.005$, so the error in the total is within $\pm 0.005 n$. If $n$ is large, you can use the normal approximation to say the error is roughly normal with zero mean and standard deviation $0.005 \cdot \frac {\sqrt n}2$