# Are random sampling with replacement methods (like bootstrapping) really representative of the population?

In many statistics courses, bootstrapping (and other random sampling with replacement methods) are suggested as ways to improve the confidence level in a statistic and improve our inference. Some even say it is a "powerful" method.

However it seems intuitively incorrect...

Say we this is our population (N = 15) : 1 - 3 - 3 - 4 - 5 - 1 - 2 - 3 - 4 - 1 - 2 - 2- 3 - 4 - 9

and this is our sample (n = 5 ) : 1 - 4 - 4 - 2 - 9

if we use the bootstrap method the 9 value has a 1/5 chance of being randomly selected each time where as in our population the 9 value only represents 1/15 of all values!

Bootstrapping can make us believe that the values we have in our sample are more frequent than they really are. Therefore it feels like there is a huge bias... Or am I missing something?

I have been looking for discussion/publications on this but I haven't found any, there seems to be a consensus on the fact that it's a powerful method, but I can't help but feel like this is a biased method that will make us overconfident of our sample.

This seems so obvious and simple that I can't imagine that all the statisticians never thought of that, so I'm guessing I'm just missing something quite elemental...

• Good question, but perhaps previously addressed here stats.stackexchange.com/questions/112147/… . Jul 1, 2020 at 15:01
• Note, one of the cited references does suggest n>20, to avoid poor performance. Jul 1, 2020 at 15:09