I'm doing an experiment comparing 10,000 rolls of 5 dice (those 5 rolls were then summed, so the population mean is 17.5) and dividing them into sets of ten with sample size N = 10. The same experiment is repeated except with 3000 rolls, but the sample size decreases to N = 3. Both experiments have 1000 samples, just with different sample sizes.
Now, I took the AVERAGE sample standard deviation for each experiment and to my great confusion, the experiment with the SMALLER sample size had a SMALLER average sample standard deviation than the experiment with the LARGER sample size. I thought this was a fluke and re-generated the random numbers only to find the same result.
In other words, this means that the sample standard deviation for each sample of N =3 is on average smaller than each sample of N = 10. How is this possible?
Intuitively, one would think that with a larger sample size the "spread" between values would be decreased because any individual improbable value would have less effect on the spread. Sort of like for the mean. While for a smaller sample size, the variables would be less "controlled" and thus on average more spread out.
Is my intuition just wrong? And is this plain to see mathematically?
FOLLOW UP: I talked to my professor and he said that a weird phenomenon happens in statistics where sample standard deviations tend to be underestimations rather than overestimations of the population standard deviation. And that as sample size goes up, the sample standard deviation goes up because it becomes "more accurate."
Is this true? And why does that happen? Is there both an intuitive explanation as well as a mathematical proof?