Calculating expected value from quantiles For probabilities $p_i=\frac{i}{10}$ where $i=1, \dots, 10$, the respective quantiles are $\tau_i$. How can I calculate an approximate expected value?

Using my answer at Expected value as a function of quantiles?, a general expression for the expectation in terms of the quantile function is
$$  \mu=\int_0^1 Q(p)\; dp
$$   in the continuous case, and that answer extends to the general case.
Looking at the approximating sums defining the integral, you can read this as the mean is the mean of the quantiles, which gives an approximation for your case as
$$ \frac{\sum_1^{10} \tau_i}{10}
$$
