variance of sample mean and sample standard deviation
I would like to know how to derive the expected variance of sample mean values and sample standard deviation.
(I am not a statistician so please forgive my sparse use of formulas.)
I have the following scenario:
1. 10 drawings from a normal distribution (M=0; SD = 1)
2a. Calculate the mean valure from these 10 drawings.
2b. Calculate the standard deviation from these 10 drawings.
3. Repeat steps 1 and 2.
Now, I would like to know the expectancy values for the standard deviation of the mean values from 2a and the standard deviation from 2b.
For the mean, it is simply SDm=((mean/sqrt(10)))^0.5
For the standard deviation, my current suggestion is:
SDsd = sqrt(0.18)^0.5
0.18 -> based on the sample variance distribution for N = 10 in a normal distribution
(Sample Variance Distribution -- from Wolfram MathWorld)
So, do my considerations make sense?
Also, in more general terms: I wonder if it is correct that the estimated standard deviation is lower for mean values than for standard deviations, since SDm > SDsd for all N.
In other words: Estimations of the standard deviation of the standard deviations are more precise than estimations of the standard deviation of sample means!?
thanks in advance for your help!