I'm calculating standard deviation for a normally distributed variable. I'm receiving data samples online, and I've noticed that right after the start (for a small sample size) standard deviation is far too away from a real. I need somehow to select the number of samples I need to wait, until I will be able to use the calculated deviation.
So, I've started my research in Excel.
I am generating normally distributed variable with $\sigma = 15$:
Measuring the population standard deviation:
And for each new sample I'm measuring the sample standard deviation for all previous samples:
And then dividing sample standard deviation by population standard deviation and obtaining this ratio for each sample size:
After refreshing it multiple times I am able to see that I need to wait something like 3000 samples before starting using sample standard deviation. But I don't really want to hardcode it, since I don't fully understand how this number can change for different dataset.
What is the right way to calculate this number?