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?

**Answer**

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.

**Attribution***Source : Link , Question Author : blackbeard , Answer Author : olooney*