# Do the set of all standardized moments of a dataset completely and uniquely define it?

I have two datasets, 'A' and 'B', comprising N measurements of one quantity, that I would like to compare to the results of a simulation, let's call this last dataset 'S'. This comparison got me thinking and I have the following questions (which are quite general I guess).

Is a dataset completely determined by its standardized moments? That is to say, given a dataset I can always calculate its zeroth order moment, the first, and so forth. If I were able to calculate them all, I'll have a function of the discrete variable 'q', the order of the standardized moment. Is this function unique? That is to say, is the following statement a mathematical truth: "if two datasets have the same standardized moments, the datasets are the same" ?

The moments define the dataset up to a permutation. Let the dataset be $x_1,\ldots,x_n$. You are given the values of $m_k = \frac{1}{n} \sum_{i=1}^n x_i^k$ for all $k$ (in fact, $k=1,\ldots,n$ suffice). Using the Newton identities, you can compute the elementary symmetric functions $\sigma_k = \sum_{\substack{S \subseteq [n] \\ |S|=k}} \prod_{i \in S} x_i$ (for example, $\sigma_1 = x_1+\cdots+x_n$ and $\sigma_2 = x_1x_2+\cdots+x_{n-1}x_n$, with $\binom{n}{2}$ terms for all possible pairs of indices). This allows you to form the polynomial $$P(x) = x^n - \sigma_1 x^{n-1} + \sigma_2 x^{n-2} - \cdots \pm \sigma_n = (x-x_1) \cdots (x-x_n),$$ whose roots are $x_1,\ldots,x_n$.