# 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?

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.