During my nuclear physics lab I was measuring decay rates of Fe59. The average was around 20 decays per second. But I did two series - in one I was measuring number of decays per 90 seconds and in the second series per 300 seconds. Standard deviation in first case was 0,54 and in second it was 0,28. How to understand this difference? Why deviations get smaller the longer measurement is done? Is it connected with some theorem in statistics?
This is equivalent to throwing a dice and finding how many sixes there are per throw.
To do the experiment find the average number of sixes per throw for a sample of 25 throws, 100 throws and 250 throws.
As the number in the sample is increased “unusual” results eg four sixes in succession, have less effect on the mean.
If these “outliers” have less effect on the mean it results in the standard deviation being less.
Use this GeoGebra statistical distribution calculator set to the binomial distribution where $n$ is the number of throws and and the probability $p=0.5$
Here is the graph for $25$ throws.
And for $100$ throws note how outliers have less effect.