My apologies is this is too rudimentary to be asked here. If it does not belong here, could someone recommend a more appropriate place to ask?
A little context. I am in a Senior Physics Lab class in college. Our first "experiment" involved rolling dice and calculating the probability of rolling each face. (The purpose is purely for practice in applying and understand statistical analysis.) I chose to roll two dice separately with the intention of testing whether the two dice are identical. I know now I should have taken more samples, but I am not able to collect more data at this time.
I originally recorded my data in trials of 20 rolls, and I took 15 trials for each die. I then calculated the counts of each face within each trial and divided by 20 to obtain an estimate of the probability in each trial. I then averaged these probabilities over all the trials and calculated the standard error (calculated as the standard deviation of the sample). However, this produced very large standard error, from about 40% to 60% for each face. Here is the data for Die 1.
Face Avg. Probability Error Percent Error 1 0.13 0.08 59.6 2 0.20 0.08 39.0 3 0.18 0.09 47.9 4 0.13 0.08 64.8 5 0.18 0.10 57.1 6 0.18 0.10 55.4
However, I discovered that by grouping the trials into larger sets, the error decreased. That is, I grouped trials 1 through 5, 6 through 10, and 11 through 15, and I calculated the probability of obtaining each face as the count of the face within the "grouped trial" divided by 100. This produced the following values for Die 1:
Face Avg. Probability Error Percent Error 1 0.13 0.03 26.6 2 0.20 0.01 5.0 3 0.18 0.04 20.0 4 0.13 0.06 43.5 5 0.18 0.06 32.9 6 0.18 0.02 11.1
Have I definitely done something wrong, or is the change in the averages and the decrease in error explainable? Would I be doing something to invalidate my results by using the larger "group trial" samples? I can provide my original data in CSV format if desired, and of course, if I am saying things that make no sense, I am open to correction.
I've found the mistake in my calculations with the groups of 20 rolls, and the means are now identical as expected. However, this hasn't resolved the question of why the error is so much larger when I group my data into a greater number of samples, which was really the question I meant to ask.
Yes, I understand that I could simply consider this problem using other means, but I'm still wondering why the error/standard deviation decreases. I think that question applies more generally than to this specific probability question. Or is the approach of trying to estimate the probability this way entirely invalid?