What is the name for the average of the largest and the smallest values in a given data set? What do you call a statistical mean that is calculated from upper and lower extremes in any given dataset?
For example, if you have a set:
{ -2, 0 , 8, 9, 1, 50, -2, 6}

The upper extreme of this set is 50 and lower extreme is -2. So, average of the extremes would be (-2 + 50 / 2) = 48/2 = 24
Is there a term for this kind of statistical mean?  
 A: It's called the midrange and while it's not the most widely used statistic in the world it does have some relevance to the uniform distribution. 
Let's introduce the order statistic notation: if have $n$ i.i.d. random variables $X_1, ..., X_n$, then the notation $X_{(i)}$ is used to refer to the $i$-th largest of the set $\{X_1, ..., X_n\}$. Thus we have:
$$ X_{(1)} ≤ X_{(2)} ≤···≤ X_{(n)} \tag{1} $$
Where $X_{(1)}$ is the minimum and $X_{(n)}$ is the maximum element. Then range and midrange are defined as:
$$ \begin{align}
  R & = X_{(n)} - X_{(1)} \tag{2} \\
  A & = \frac{X_{(1)} + X_{(n)}}{2} \tag{3} \\
  \end{align}
$$
These formulas are taken from CRC Standard Probability and Statistics Tables and Formulae, section 4.6.6.
If $X_i$ is assumed to have a uniform distribution $X_i \sim U(\alpha, \beta)$, where $\alpha$ and $\beta$ are the lower and upper bounds respectively, then we can give the MLE estimates in terms of these formulas:
$$ 
 \begin{align}
  \hat{\alpha} & = X_{(1)} \tag{4} \\
  \hat{\beta} & = X_{(n)} \tag{5}
 \end{align}
$$
The mean of the resulting distribution is the same as the midrange:
$$ 
 \begin{align}
  \mu & = A = \frac{X_{(1)} + X_{(n)}}{2} \tag{6} \\
 \end{align}
$$
This is probably the only use for this particular statistic. 
