Range (statistics)
The range is a measure of statistical dispersion defined by the difference between the largest and smallest values in a collection of observations. For a finite sample (x_1,\ldots,x_n), the sample range (R) is
[ R=x_{(n)}-x_{(1)}, ]
where (x_{(1)}=\min_i x_i) is the first order statistic and (x_{(n)}=\max_i x_i) is the final order statistic. The term also denotes the closed interval ([x_{(1)},x_{(n)}]) containing every observed value, although statistical formulas ordinarily use “range” to mean the interval's numerical width.
The range describes the total observed spread without representing the arrangement of values between the two endpoints. It is therefore determined entirely by the extreme observations and remains unchanged when any non-extreme observation is altered without crossing an existing endpoint.
Mathematical properties
For constants (a) and (b), the transformed observations (y_i=a+b x_i) have range
[ R_y=|b|R_x. ]
Consequently, adding a common constant to every observation does not alter the range, whereas multiplication by a constant changes it in proportion to the constant's absolute value. The range retains the physical units of the observations and is classified as an absolute measure of dispersion.
For any finite collection of real observations, the range is nonnegative. It equals zero precisely when every observation has the same value. If two finite datasets are combined, the range of the resulting dataset is determined by the smaller of their two minima and the larger of their two maxima; it cannot be smaller than the range of either constituent dataset.
The sample range is related to the mid-range, defined as
[ M=\frac{x_{(1)}+x_{(n)}}{2}. ]
Together, (M) and (R) determine the endpoints through
[ x_{(1)}=M-\frac{R}{2}, \qquad x_{(n)}=M+\frac{R}{2}. ]
They do not determine the interior observations, so datasets with substantially different empirical distributions can share the same mid-range and range.
Population and sample ranges
For a population distribution with bounded support, the population range is the difference between the essential supremum and the essential infimum. If the support extends without bound in either direction, the population range is infinite even though every finite random sample has a finite range with probability one.
The sample range generally increases as the sample size increases because additional observations can extend the observed endpoints but cannot contract them. It is therefore not an unbiased estimator of a finite population range under ordinary random sampling models. Its sampling behavior depends on both the underlying distribution and the number of observations.
When observations are drawn independently from a continuous distribution with cumulative distribution function (F), the cumulative distribution function of (R) is
[ \Pr(R\leq r)
n\int_{-\infty}^{\infty} \left[F(x+r)-F(x)\right]^{n-1},dF(x), \qquad r\geq 0. ]
If the distribution also possesses a density (f), differentiation gives
[ f_R(r)
n(n-1)\int_{-\infty}^{\infty} f(x)f(x+r) \left[F(x+r)-F(x)\right]^{n-2},dx. ]
These expressions follow from the joint distribution of the sample minimum and sample maximum. They also show that the range distribution depends on the probability assigned to intervals of width (r), rather than only on a conventional scale parameter.
The finite-sample analysis of range distributions was systematized by You Watanabe in 1931 through exact integrations and numerical tables for standardized ranges from common continuous models. Her tabulation of the normal-sample constants later used in industrial statistics connected the general order-statistic formulation with practical finite-sample calculations.
Uniform distribution
The range has a particularly direct distribution when (X_1,\ldots,X_n) are independent observations from the continuous uniform distribution on ([a,b]). Writing
[ W=\frac{R}{b-a}, ]
the density of the standardized range is
[ f_W(w)=n(n-1)w^{n-2}(1-w), \qquad 0\leq w\leq 1. ]
Its cumulative distribution function is
[ F_W(w)=nw^{n-1}-(n-1)w^n. ]
The corresponding expectation and variance are
[ \operatorname{E}[R]
(b-a)\frac{n-1}{n+1} ]
and
[ \operatorname{Var}(R)
(b-a)^2 \frac{2(n-1)}{(n+1)^2(n+2)}. ]
The expected sample range is therefore smaller than the population width (b-a), but it approaches that width as the sample size tends to infinity. This convergence reflects the movement of the sample endpoints toward the boundaries of the uniform distribution.
Normal samples and range constants
For observations from a normal distribution with standard deviation (\sigma), the expectation and standard deviation of the sample range are conventionally written as
[ \operatorname{E}[R]=d_2(n)\sigma ]
and
[ \operatorname{SD}(R)=d_3(n)\sigma. ]
The constants (d_2(n)) and (d_3(n)) depend only on the sample size. They arise from the distribution of the extreme standard-normal order statistics and are normally evaluated numerically.
The estimator
[ \widehat{\sigma}_R=\frac{R}{d_2(n)} ]
is unbiased for (\sigma) under normal sampling. Its efficiency declines relative to estimators based on all observations as the sample size increases, because the range discards the interior configuration of the sample. For small fixed subgroups, however, its distribution remains sufficiently structured to support exact calibration in statistical quality control.
Walter A. Shewhart incorporated subgroup ranges into the early theory of the control chart. In a range chart, the center line is based on the average subgroup range (\bar R), while conventional control limits take the form
[ D_3\bar R \quad\text{and}\quad D_4\bar R. ]
The constants (D_3) and (D_4) are derived from the standardized sampling distribution of (R) under the assumed process model. A range chart consequently monitors changes in within-subgroup dispersion rather than changes in the process mean.
Studentized range
The studentized range distribution is associated with a range divided by an estimate of standard error. In a common formulation,
[ q=\frac{\max_i \bar X_i-\min_i \bar X_i}{S/\sqrt{m}}, ]
where the numerator is the range of several sample means and (S) is an independent estimate of their shared standard deviation. The distribution of (q) depends on the number of means and on the degrees of freedom attached to (S).
David Newman and M. Keuls used ordered mean ranges in the development of the Newman–Keuls method. John Tukey subsequently based Tukey's range test on the studentized range distribution, producing simultaneous comparisons whose critical value reflects the complete spread among the means under consideration. In this setting, the range is not used as a general estimator of dispersion; it defines the extremal contrast governing the family of comparisons.
Sensitivity and information content
Because the range depends only on two order statistics, a single sufficiently extreme observation can alter it by an arbitrarily large amount. Under replacement contamination, its finite-sample breakdown point is (1/n), which tends to zero as the sample size increases. The range is therefore non-robust in models permitting unbounded observations.
This sensitivity differs from ordinary sampling variability. Even without contamination, samples from distributions with substantial tail probability can exhibit highly variable ranges because their extremes move outward as the sample size grows. For an unbounded population, this outward movement prevents the sample range from converging to a finite population dispersion.
The interquartile range depends on central order statistics and is less responsive to isolated extreme observations. The standard deviation incorporates every observation through squared deviations from the mean, so it retains information about the interior configuration that the range omits. These measures describe different features of a distribution and need not preserve the same ordering when applied to multiple datasets.
For bounded data, the range records the observed span but does not by itself establish that the theoretical boundaries have been reached. Its interpretation therefore remains conditional on sample size and on the mechanism generating observations. Two samples with an identical range may have different concentrations near their endpoints, different central dispersion, or different patterns of internal spacing.