Range (statistics)

rangerangingsample rangemoving rangeranged order
In statistics, the range of a set of data is the difference between the largest and smallest values.wikipedia
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Statistical dispersion

dispersionvariabilityspread
The range is the size of the smallest interval (statistics) which contains all the data and provides an indication of statistical dispersion.
Range

Descriptive statistics

descriptivedescriptive statisticstatistics
However, in descriptive statistics, this concept of range has a more complex meaning.
Univariate analysis involves describing the distribution of a single variable, including its central tendency (including the mean, median, and mode) and dispersion (including the range and quartiles of the data-set, and measures of spread such as the variance and standard deviation).

Order statistic

order statisticsorderedth-smallest of items
The range is a simple function of the sample maximum and minimum and these are specific examples of order statistics.
The sample range is the difference between the maximum and minimum.

Sample maximum and minimum

sample maximumsample minimumMaximum
The range is a simple function of the sample maximum and minimum and these are specific examples of order statistics.
In addition to being a component of every statistic that uses all elements of the sample, the sample extrema are important parts of the range, a measure of dispersion, and mid-range, a measure of location.

L-estimator

L-estimation
In particular, the range is a linear function of order statistics, which brings it into the scope of L-estimation.
A more detailed list of examples includes: with a single point, the maximum, the minimum, or any single order statistic or quantile; with one or two points, the median; with two points, the mid-range, the range, the midsummary (trimmed mid-range, including the midhinge), and the trimmed range (including the interquartile range and interdecile range); with three points, the trimean; with a fixed fraction of the points, the trimmed mean (including interquartile mean) and the Winsorized mean; with all points, the mean.

Interquartile range

inter-quartile rangebelowinterquartile
Interquartile range
It is a trimmed estimator, defined as the 25% trimmed range, and is a commonly used robust measure of scale.

Statistics

statisticalstatistical analysisstatistician
In statistics, the range of a set of data is the difference between the largest and smallest values.

Interval estimation

interval estimateintervalinterval (statistics)
The range is the size of the smallest interval (statistics) which contains all the data and provides an indication of statistical dispersion.

Independent and identically distributed random variables

independent and identically distributedi.i.d.iid
For n independent and identically distributed continuous random variables X 1, X 2, ..., X n with cumulative distribution function G(x) and probability density function g(x). Let T denote the range of a sample of size n from a population with distribution function G(x).

Cumulative distribution function

distribution functionCDFcumulative probability distribution function
For n independent and identically distributed continuous random variables X 1, X 2, ..., X n with cumulative distribution function G(x) and probability density function g(x). Let T denote the range of a sample of size n from a population with distribution function G(x). For n independent and identically distributed discrete random variables X 1, X 2, ..., X n with cumulative distribution function G(x) and probability mass function g(x) the range of the X i is the range of a sample of size n from a population with distribution function G(x). We can assume without loss of generality that the support of each X i is {1,2,3,...,N} where N is a positive integer or infinity.

Probability density function

probability densitydensity functiondensity
For n independent and identically distributed continuous random variables X 1, X 2, ..., X n with cumulative distribution function G(x) and probability density function g(x). Let T denote the range of a sample of size n from a population with distribution function G(x).

Emil Julius Gumbel

GumbelGumbel, Emil JuliusE[mil] J. Gumbel
Gumbel notes that the "beauty of this formula is completely marred by the facts that, in general, we cannot express G(x + t) by G(x), and that the numerical integration is lengthy and tiresome."

Bessel function

modified Bessel functionspherical Bessel functionBessel
For more general distributions the asymptotic distribution can be expressed as a Bessel function.

Normal distribution

normally distributednormalGaussian
In the case where each of the X i has a standard normal distribution, the mean range is given by

Probability mass function

mass functionprobability massmass
For n independent and identically distributed discrete random variables X 1, X 2, ..., X n with cumulative distribution function G(x) and probability mass function g(x) the range of the X i is the range of a sample of size n from a population with distribution function G(x). We can assume without loss of generality that the support of each X i is {1,2,3,...,N} where N is a positive integer or infinity.

Without loss of generality

WLOGloss of generalityw.l.o.g.
For n independent and identically distributed discrete random variables X 1, X 2, ..., X n with cumulative distribution function G(x) and probability mass function g(x) the range of the X i is the range of a sample of size n from a population with distribution function G(x). We can assume without loss of generality that the support of each X i is {1,2,3,...,N} where N is a positive integer or infinity.

Support (mathematics)

supportcompact supportcompactly supported
For n independent and identically distributed discrete random variables X 1, X 2, ..., X n with cumulative distribution function G(x) and probability mass function g(x) the range of the X i is the range of a sample of size n from a population with distribution function G(x). We can assume without loss of generality that the support of each X i is {1,2,3,...,N} where N is a positive integer or infinity.

Discrete uniform distribution

uniform distributionuniformly distributeduniformly at random
If we suppose that g(x) = 1/N, the discrete uniform distribution for all x, then we find

Reference range

reference valuesreference rangescut-off
In health-related fields, a reference range or reference interval is the range of values that is deemed normal for a physiologic measurement in healthy persons (for example, the amount of creatinine in the blood, or the partial pressure of oxygen).

Qualitative variation

M2Margalef index
Several are standard statistics that are used elsewhere - range, standard deviation, variance, mean deviation, coefficient of variation, median absolute deviation, interquartile range and quartile deviation.

Tetraplasandra

ohe mauka
The range of variation in Tetraplasandra is unusually large for a genus in Araliaceae Most of the species were originally described in other genera which were later merged with Tetraplasandra.

Rescaled range

R/S analysis
The rescaled range is calculated from dividing the range of the values exhibited in a portion of the time series by the standard deviation of the values over the same portion of the time series.

HSL and HSV

HSVHSLHSV color wheel
Because these definitions of saturation—in which very dark (in both models) or very light (in HSL) near-neutral colors are considered fully saturated (for instance, from the bottom right in the sliced HSL cylinder or from the top right)—conflict with the intuitive notion of color purity, often a conic or biconic solid is drawn instead, with what this article calls chroma as its radial dimension (equal to the range of the RGB values), instead of saturation (where the saturation is equal to the chroma over the maximum chroma in that slice of the (bi)cone).

Mid-range

midsummarymidrangehalf-range
The mid-range is the midpoint of the range; as such, it is a measure of central tendency.