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?

  • 12
    $\begingroup$ It's the "midrange". $\endgroup$
    – jbowman
    Jul 18, 2019 at 22:07

1 Answer 1


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.

  • 3
    $\begingroup$ Historicaly, the mean air temperature of a day was given as the midrange. $\endgroup$
    – Maxter
    Jul 19, 2019 at 17:05

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