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Bootstrapping approach is based on the assumption that the sample is the population. Then, a resampling of the sample is done, replacing the individuals of the population by chance. The number of the resampled individuals is the same as the 'real' sample. So, how is the new sample different from the original one? I don't have a huge background in statistic so please be plain.

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@Glen_b has provided a good answer that addresses the main point. I just thought I'd add that the number of resampled observations does not have to be the same as the number of elements in the original sample. For most of the underlying asymptotic theory to work, all that is required (from memory) is that the number of resampled observations and the number of observations in the original sample grow at the same rate. –  Colin T Bowers Feb 6 at 9:43
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So, how is the new sample different from the original one? I don't have a huge background in statistic so please be plain.

Sampling is with replacement.

Imagine you have a sample of size 30. Take a 30-sided die and write each value on a face.

Roll the die 30 times, recording the values. That's one pseudo-sample.

A value from the original sample has a bit over 60% (asymptotically, $1-1/e$) chance of turning up in the new sample, and some of the observations will appear two or three times, or sometimes more, which makes up for the ones that didn't show up.

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