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Dividing data of a given distribution

Having a dataset with distribution $p$ (e.g. uniform or normal), if we divide the dataset into $n$ parts with equal size, is it valid to say that each part still has distribution $p$?
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Splitting a dataset with a given distribution?

Having a dataset with a given distribution $p$ (e.g. uniform or normal), if the dataset is randomly split into $n$ sub-datasets with equal size, is it valid to say that each sub-dataset still has ...