Consider an $n$ qubit state $|\psi\rangle$. Let us say we have the following upper bound on the total variation distance between the distribution generated by measuring $|\psi\rangle$ in the standard basis and the uniform distribution.
$$\sum_{x \in \{0, 1\}^{n}} \left||\langle x | \psi \rangle|^{2} - \frac{1}{2^{n}} \right| \leq a,$$
and, additionally,
$$ \left||\langle x | \psi \rangle|^{2} - \frac{1}{2^{n}} \right| \leq \frac{a}{2^{n}},$$
for every $x$.
Now, let us consider a Pauli observable $O$ as follows:
$$O = \sum_{\sigma_i\in\{I, X, Y, Z\}^{\otimes n}} w_{\sigma_i} \sigma_i. $$
Does an upper bound on the total variation distance imply an upper bound on the following quantity?
$$\left|\langle \psi|O |\psi \rangle - \frac{\text{Tr}(O)}{2^{n}} \right|?$$
My intuition is that a change of basis seems to be at the heart of the question -- we are changing from the standard basis to the Pauli basis. How does the total variation distance change with this change of basis?