Let's say I have an array of numbers that have a specific order, say:
$$x_1, x_2, x_3, \cdots x_n$$
Problem: How do I calculate the mean of $\Delta x$, where $\Delta x = x_{n+1}-x_n$ ?
At first glance it might seem trivial: $$\langle \Delta x \rangle = \frac1{n-1}\sum_{i=0}^n (x_{i+1}-x_i)$$ But let's look at an example. Suppose that I have the following four numbers: $$1,5,8,11$$ Using the equation above would mean: $$\langle \Delta x \rangle = \frac13\big[(5-1)+(8-5)+(11-8)\big] = \frac1310 \approx 3.33$$ But I could also sum the terms: $$\frac{11-5}{2} \quad ; \frac{8-1}{2}$$ Which would mean: $$\frac15\bigg[(5-1)+(8-5)+(11-8)+3+\frac72\bigg]=\frac{33}{10} = 3.3$$ Which is a different result. For me it seems that the first result is correct, because it agrees with the definition of mean.
Question: Which one is correct, and why does taking the two extra terms I proposed gives an incorrect result?
Your first attempt is correct.
Note that you can simplify the process by telescoping your sum $$ \sum _{i=0}^{n-1} (x_{i+1}-x_i) = x_n-x_0$$ Thus your mean is simply $$ \frac {x_n-x_0}{n-1}$$