# 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 the "midrange". Jul 18, 2019 at 22:07

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.

• Historicaly, the mean air temperature of a day was given as the midrange. Jul 19, 2019 at 17:05