I have sets $A = \left\{a_{1},a_{2},a_{3},\dots,a_{n} \right\}$ and $B = \left\{b_{1},b_{2},b_{3},\dots,b_{n} \right\}$. If they are time-series sorted by their indices, I can take the differences, $$ d = \frac{\Delta A}{\Delta B} = \left\{\frac{a_2-a_1}{b_2-b_1},\frac{a_3-a_2}{b_3-b_2}, \dots, \frac{a_n-a_{n-1}}{b_n-b_{n-1}}\right\} $$ and take the mean finite difference $\bar{d}$ to approximate the derivative $\frac{dA(t)}{dB(t)}$.
If they are not sorted by time (i.e., they are just sequences but not time-series), can I take all combinations of differences in the numerator and denominator, take their ratio, and finally their mean to approximate $\bar{d}$? That is, can we say: $$ \bar{d} \approx \frac{1}{n}\sum_{i\neq j}\frac{a_i-a_j}{b_i-b_j} $$
It will be awesome if someone could suggest an approximation. This is needed in my biology research. Thanks.
