I know that the standard representation for the average of a data set: $$ \bar{x} = \frac{1}{N} \sum_{i}^N{x_i} $$
I have also ran into an average denoted as $\langle x \rangle$. This notation is frequently used in physics, and I understand it to be a continuous average over either time or space:
$$ \langle x \rangle = \frac{1}{b-a} \int_a^b{f(x)} $$
Where $f(x)$ can be a function of any coordinates of space and/or time.
From this there is also the relation as $N \to \infty$ and $\Delta(x_i, x_{\pm i}) \to 0 $:
$$ \bar{x} = \langle x \rangle = \frac{1}{\infty}\sum_i^\infty{x_i} = \frac{1}{b-a} \int_a^bf(x) $$
This indicates that given enough samples, it should be the case that $\bar{x}=\langle x \rangle$.
Am I correct in this understanding that $\bar{x}$ represents a discrete average and $\langle x \rangle$ represents a continuous average, or is there something else to this?
For the average of p(t) over T, without calculus I would propose:
$$P = {\left.\overline{p(t)}\right|_0^T}$$
in preference of
$$P = \frac{1}{T}\int_0^T p(t)\ d t.$$
The reason being that the integration sign is intimidating for people without a calculus background.
Your comments would be most welcome.