Is there a phenomenon like this? It's like the mean of every part increases, but the overall mean decrease; or the mean of a group is larger in every part but is smaller in all samples; or something like this.
I remember there is a name for this phenomenon like this (it may not be accurate) but my impression is very vague and I cannot remember it clearly.

You are thinking of (one version of) Simpson's paradox.  You can find many threads about it on the site.
