Apologies if this is a little confusing. If anything needs clarifying, I'll edit the question as best I can. Thanks.
If we take a sample of, say, 100 people, we could measure their heights, and find the mean and standard deviation. We can calculate an estimate of the standard error from this single sample to find where the population mean is likely to lie.
If we repeated this sampling and measured further samples of 100 people from the population, then finding the means for each sample would give a sampling distributions of means. The mean of this sampling distribution estimates the population mean, while the standard deviation of this sampling distribution also estimates the standard error.
This all seems fairly clear.
Now, consider a large number of trials, e.g. 20000 trials of 100 coin flips, where the number of heads in each trial is recorded and the fraction of heads obtained in each trial found. To me, it seems less obvious as to what constitutes a sample here.
Would each one of these 20000 trials be considered as a single sample, so that we have 20000 samples?
If that's right, then when we average the fractions of heads found over all 20000 trials, then — just as for the height example — the mean is an estimate of the population mean, and the standard deviation would be an estimate of the standard error.
The alternative I'm thinking of would be to treat each set of 100 coin flips as a single data point, which would give a single large sample of 20000 measurements.
Treating this data as 20000 individual samples does seem like it should be correct since it's analogous to the height example. However, it doesn't seem immediately intuitive to me. Maybe this is because each trial results in a single fraction value, instead of dealing with the 100 individual flips of each trial. So, thinking of the result of each trial as a individual sample mean seems less obvious here, as the trial effectively yields a single value. On the other hand, I can see that the fraction of heads is effectively a mean value, where we code heads as 1, tails as 0 and then divide by the total number of flips in a trial.
What confuses me further is if we then decide to work with the raw number of heads found in each trial instead of the fraction of heads. The total number of heads in a trial is a single value, rather than a mean. However, my thinking is that the same would be true: the mean of the 20000 raw numbers of heads would be an estimate of the population mean, and that the standard deviation would estimate the standard error of the mean. I'm not sure if or why this would be the case though.
Edit: Really, I think what I'm trying to understand about this:
If I take repeated trials of a fixed number of flips and compile the fraction of heads, do these give a distribution of sample means? I think from Michael Chernick's answer that this is correct.
Next, if we look at the distribution of the raw scores of heads obtained in the repeated trials of a fixed number of flips, can this be again considered as a distribution of sample means? This is exactly the same as in 1, except the measured values are multiplied by the sample size. Because of this, I assume the answer is yes, but it just seems a little dissonant. When you reframe the data like that, you're using a total value, rather than the mean number of heads (the mean being equivalent to the fraction of heads).