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In
statistics Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, the mid-range or mid-extreme is a measure of
central tendency In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications in ...
of a sample defined as the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The coll ...
of the maximum and minimum values of the data set: :M=\frac. The mid-range is closely related to the range, a measure of
statistical dispersion In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquarti ...
defined as the difference between maximum and minimum values. The two measures are complementary in sense that if one knows the mid-range and the range, one can find the sample maximum and minimum values. The mid-range is rarely used in practical statistical analysis, as it lacks efficiency as an estimator for most
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a varia ...
s of interest, because it ignores all intermediate points, and lacks
robustness Robustness is the property of being strong and healthy in constitution. When it is transposed into a system, it refers to the ability of tolerating perturbations that might affect the system’s functional body. In the same line ''robustness'' ca ...
, as outliers change it significantly. Indeed, for many distributions it is one of the least efficient and least robust statistics. However, it finds some use in special cases: it is the maximally efficient estimator for the center of a uniform distribution, trimmed mid-ranges address robustness, and as an
L-estimator In statistics, an L-estimator is an estimator which is a linear combination of order statistics of the measurements (which is also called an L-statistic). This can be as little as a single point, as in the median (of an odd number of values), or ...
, it is simple to understand and compute.


Robustness

The midrange is highly sensitive to outliers and ignores all but two data points. It is therefore a very non-
robust statistic Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal distribution, normal. Robust Statistics, statistical methods have been developed ...
, having a breakdown point of 0, meaning that a single observation can change it arbitrarily. Further, it is highly influenced by outliers: increasing the sample maximum or decreasing the sample minimum by ''x'' changes the mid-range by x/2, while it changes the sample mean, which also has breakdown point of 0, by only x/n. It is thus of little use in practical statistics, unless outliers are already handled. A trimmed midrange is known as a – the ''n''% trimmed midrange is the average of the ''n''% and (100−''n'')% percentiles, and is more robust, having a breakdown point of ''n''%. In the middle of these is the
midhinge In statistics, the midhinge is the average of the first and third quartiles and is thus a measure of location. Equivalently, it is the 25% trimmed mid-range or 25% midsummary; it is an L-estimator. : \operatorname(X) = \overline = \frac = \frac = ...
, which is the 25% midsummary. The
median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...
can be interpreted as the fully trimmed (50%) mid-range; this accords with the convention that the median of an even number of points is the mean of the two middle points. These trimmed midranges are also of interest as
descriptive statistics A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features from a collection of information, while descriptive statistics (in the mass noun sense) is the process of using and an ...
or as
L-estimator In statistics, an L-estimator is an estimator which is a linear combination of order statistics of the measurements (which is also called an L-statistic). This can be as little as a single point, as in the median (of an odd number of values), or ...
s of central location or
skewness In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For a unimodal ...
: differences of midsummaries, such as midhinge minus the median, give measures of skewness at different points in the tail.


Efficiency

Despite its drawbacks, in some cases it is useful: the midrange is a highly efficient
estimator In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the ...
of μ, given a small sample of a sufficiently platykurtic distribution, but it is inefficient for mesokurtic distributions, such as the normal. For example, for a
continuous uniform distribution In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies betwe ...
with unknown maximum and minimum, the mid-range is the uniformly minimum-variance unbiased estimator (UMVU) estimator for the mean. The
sample maximum In statistics, the sample maximum and sample minimum, also called the largest observation and smallest observation, are the values of the greatest and least elements of a sample. They are basic summary statistics, used in descriptive statistics ...
and sample minimum, together with sample size, are a sufficient statistic for the population maximum and minimum – the distribution of other samples, conditional on a given maximum and minimum, is just the uniform distribution between the maximum and minimum and thus add no information. See
German tank problem In the statistical theory of estimation, the German tank problem consists of estimating the maximum of a discrete uniform distribution from sampling without replacement. In simple terms, suppose there exists an unknown number of items which ar ...
for further discussion. Thus the mid-range, which is an unbiased and sufficient estimator of the population mean, is in fact the UMVU: using the sample mean just adds noise based on the uninformative distribution of points within this range. Conversely, for the normal distribution, the sample mean is the UMVU estimator of the mean. Thus for platykurtic distributions, which can often be thought of as between a uniform distribution and a normal distribution, the informativeness of the middle sample points versus the extrema values varies from "equal" for normal to "uninformative" for uniform, and for different distributions, one or the other (or some combination thereof) may be most efficient. A robust analog is the
trimean In statistics the trimean (TM), or Tukey's trimean, is a measure of a probability distribution's location defined as a weighted average of the distribution's median and its two quartiles: : TM= \frac This is equivalent to the average of the ...
, which averages the midhinge (25% trimmed mid-range) and median.


Small samples

For small sample sizes (''n'' from 4 to 20) drawn from a sufficiently platykurtic distribution (negative
excess kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurt ...
, defined as γ2 = (μ4/(μ2)²) − 3), the mid-range is an efficient estimator of the mean ''μ''. The following table summarizes empirical data comparing three estimators of the mean for distributions of varied kurtosis; the modified mean is the
truncated mean A truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median. It involves the calculation of the mean after discarding given parts of a probability distribution or sample at the high and low end, a ...
, where the maximum and minimum are eliminated. For ''n'' = 1 or 2, the midrange and the mean are equal (and coincide with the median), and are most efficient for all distributions. For ''n'' = 3, the modified mean is the median, and instead the mean is the most efficient measure of central tendency for values of ''γ''2 from 2.0 to 6.0 as well as from −0.8 to 2.0.


Sampling properties

For a sample of size ''n'' from the standard normal distribution, the mid-range ''M'' is unbiased, and has a variance given by: :\operatorname(M)=\frac. For a sample of size ''n'' from the standard
Laplace distribution In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponen ...
, the mid-range ''M'' is unbiased, and has a variance given by: :\operatorname(M)=\frac and, in particular, the variance does not decrease to zero as the sample size grows. For a sample of size ''n'' from a zero-centred uniform distribution, the mid-range ''M'' is unbiased, ''nM'' has an
asymptotic distribution In mathematics and statistics, an asymptotic distribution is a probability distribution that is in a sense the "limiting" distribution of a sequence of distributions. One of the main uses of the idea of an asymptotic distribution is in providing ap ...
which is a
Laplace distribution In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponen ...
.


Deviation

While the mean of a set of values minimizes the sum of squares of deviations and the
median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...
minimizes the
average absolute deviation The average absolute deviation (AAD) of a data set is the average of the absolute deviations from a central point. It is a summary statistic of statistical dispersion or variability. In the general form, the central point can be a mean, median, ...
, the midrange minimizes the maximum deviation (defined as \max\left, x_i-m\): it is a solution to a variational problem.


See also

*
Range (statistics) In statistics, the range of a set of data is the difference between the largest and smallest values, the result of subtracting the sample maximum and minimum. It is expressed in the same units as the data. In descriptive statistics, range is ...
*
Midhinge In statistics, the midhinge is the average of the first and third quartiles and is thus a measure of location. Equivalently, it is the 25% trimmed mid-range or 25% midsummary; it is an L-estimator. : \operatorname(X) = \overline = \frac = \frac = ...


References

* * * {{DEFAULTSORT:Mid-Range Means Summary statistics