Not sure if its the optimum method but ... f[n_] := Table[Mean[RandomSample[data, 7]], {n}];

ListPlot[f[100]]

Slightly more efficient, I suspect:

f[n_]:=Mean /@ Partition[RandomChoice[data, 7 n], 7]

Also note (!) the distinction that this needs to be RandomChoice rather than RandomSample. RandomChoice allows random selection from the list of data without using any of the data more than once. RandomSample allows for choosing a random datum more than once. I assume that this is actually what you want.

If you actually want RandomSample then the original approach posted above by Melvyn would be better, but the lenght of data must be greater than or equal to 7.

Wow! Thank you! Cool code! I think you accidentally wrote it backwards? RandomChoice - samples "with replacement", meaning you can draw the same list item more than once in a sample, whereas RandomSample will not (let you). But for (my) statistical purposes, either is fine as the sample is less than 5% of the population (the data set,) and the data set is greater than 30 items.

Even more efficient would be to use:

f[n_] := Mean /@ RandomChoice[data, {n , 7}]