If you try to measure and do it really poorly, if you do it enough times, the random walk of the errors tends to average out and the average is still a decent data point.
The more errors you have will cause a wider dispersion of the data and the standard deviation will be larger.
However, as you well know, with more data points, the standard deviation narrows.
A narrowing standard deviation means higher confidence levels and greater precision of the average.
Key word there is “random”.