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Range (statistics)

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From Wikipedia, the free encyclopedia

Concept in statistics

Not to be confused withMid-range.

For other uses, seeRange (disambiguation) § Mathematics.

Indescriptive statistics, therange of a set of data is size of the narrowestinterval which contains all the data.It is calculated as the difference between the largest and smallest values (also known as thesample maximum and minimum).^[1] It is expressed in the sameunits as the data.

The range provides an indication ofstatistical dispersion. Closely related alternative measures are theInterdecile range and theInterquartile range.

Range of continuous IID random variables

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Fornindependent and identically distributed continuous random variablesX₁,X₂, ...,X_n with thecumulative distribution function G(x) and aprobability density function g(x), let T denote the range of them, that is, T= max(X₁,X₂, ...,X_n)-min(X₁,X₂, ...,X_n).

Distribution

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The range, T, has the cumulative distribution function^[2]^[3]

F(t)=n\int _{-\infty }^{\infty }g(x)[G(x+t)-G(x)]^{n-1}\,{\text{d}}x.

Gumbel notes that the "beauty of this formula is completely marred by the facts that, in general, we cannot expressG(x + t) byG(x), and that the numerical integration is lengthy and tiresome."^[2]^: 385

If the distribution of eachX_i is limited to the right (or left) then the asymptotic distribution of the range is equal to the asymptotic distribution of the largest (smallest) value. For more general distributions the asymptotic distribution can be expressed as aBessel function.^[2]

Moments

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The mean range is given by^[4]

n\int _{0}^{1}x(G)[G^{n-1}-(1-G)^{n-1}]\,{\text{d}}G

wherex(G) is the inverse function. In the case where each of theX_i has astandard normal distribution, the mean range is given by^[5]

\int _{-\infty }^{\infty }(1-(1-\Phi (x))^{n}-\Phi (x)^{n})\,{\text{d}}x.

Derivation of the distribution

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Please note that the following is an informal derivation of the result. It is a bit loose with the calculation of the probabilities.

Let $m, M {\displaystyle m,M}$ denote respectively the min and max of the random variables $X_{1}\dots X_{n}$ .

The event that the range is smaller than $T {\displaystyle T}$ can be decomposed into smaller events according to:

the index of the minimum value
and the value $x {\displaystyle x}$ of the minimum.

For a given index $i {\displaystyle i}$ and minimum value $x {\displaystyle x}$ , the probability of the joint event:

$X_{i}$ is the minimum,
and $X_{i}=x$ ,
and the range is smaller than $T {\displaystyle T}$ ,

is: $g(x)\left[G(x+T)-G(x)\right]^{n-1}$ Summing over the indices and integrating over $x {\displaystyle x}$ yields the total probability of the event: "the range is smaller than $T {\displaystyle T}$ " which is exactly the cumulative density function of the range: $F(t)=n\int _{-\infty }^{\infty }g(x)\left[G(t+x)-G(x)\right]^{n-1}\,{\text{d}}x$ which concludes the proof.

The range in other models

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Outside of the IID case with continuous random variables, other cases have explicit formulas. These cases are of marginal interest.

non-IID continuous random variables.^[3]
Discrete variables supported on $\mathbb {N}$ .^[6]^[7] A key difficulty for discrete variables is that the range is discrete. This makes the derivation of the formula requirecombinatorics.

Related quantities

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The range is a specific example oforder statistics. In particular, the range is a linear function of order statistics, which brings it into the scope ofL-estimation.

References

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^George Woodbury (2001).An Introduction to Statistics. Cengage Learning. p. 74.ISBN 0534377556.
^^a ^b ^cE. J. Gumbel (1947)."The Distribution of the Range".The Annals of Mathematical Statistics.18 (3):384–412.doi:10.1214/aoms/1177730387.JSTOR 2235736.
^^a ^bTsimashenka, I.; Knottenbelt, W.;Harrison, P. (2012). "Controlling Variability in Split-Merge Systems".Analytical and Stochastic Modeling Techniques and Applications(PDF). Lecture Notes in Computer Science. Vol. 7314. p. 165.doi:10.1007/978-3-642-30782-9_12.ISBN 978-3-642-30781-2.
^H. O. Hartley;H. A. David (1954)."Universal Bounds for Mean Range and Extreme Observation".The Annals of Mathematical Statistics.25 (1):85–99.doi:10.1214/aoms/1177728848.JSTOR 2236514.
^L. H. C. Tippett (1925). "On the Extreme Individuals and the Range of Samples Taken from a Normal Population".Biometrika.17 (3/4):364–387.doi:10.1093/biomet/17.3-4.364.JSTOR 2332087.
^Evans, D. L.; Leemis, L. M.; Drew, J. H. (2006). "The Distribution of Order Statistics for Discrete Random Variables with Applications to Bootstrapping".INFORMS Journal on Computing.18:19–30.doi:10.1287/ijoc.1040.0105.
^Irving W. Burr (1955)."Calculation of Exact Sampling Distribution of Ranges from a Discrete Population".The Annals of Mathematical Statistics.26 (3):530–532.doi:10.1214/aoms/1177728500.JSTOR 2236482.

Statistics

Descriptive statistics

Continuous data

Center	Mean Arithmetic Arithmetic-Geometric Contraharmonic Cubic Generalized/power Geometric Harmonic Heronian Heinz Lehmer Median Mode
Dispersion	Average absolute deviation Coefficient of variation Interquartile range Percentile Range Standard deviation Variance
Shape	Central limit theorem Moments Kurtosis L-moments Skewness

Count data

Index of dispersion

Summary tables

Dependence

Graphics

Data collection

Study design	Effect size Missing data Optimal design Population Replication Sample size determination Statistic Statistical power
Survey methodology	Sampling Cluster Stratified Opinion poll Questionnaire Standard error
Controlled experiments	Blocking Factorial experiment Interaction Random assignment Randomized controlled trial Randomized experiment Scientific control
Adaptive designs	Adaptive clinical trial Stochastic approximation Up-and-down designs
Observational studies	Cohort study Cross-sectional study Natural experiment Quasi-experiment

Statistical inference

Statistical theory

Frequentist inference

Point estimation	Estimating equations Maximum likelihood Method of moments M-estimator Minimum distance Unbiased estimators Mean-unbiased minimum-variance Rao–Blackwellization Lehmann–Scheffé theorem Median unbiased Plug-in
Interval estimation	Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife
Testing hypotheses	1- & 2-tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons
Parametric tests	Likelihood-ratio Score/Lagrange multiplier Wald

Specific tests

Z-test(normal) Student'st-test F-test
Goodness of fit	Chi-squared G-test Kolmogorov–Smirnov Anderson–Darling Lilliefors Jarque–Bera Normality(Shapiro–Wilk) Likelihood-ratio test Model selection Cross validation AIC BIC
Rank statistics	Sign Sample median Signed rank(Wilcoxon) Hodges–Lehmann estimator Rank sum(Mann–Whitney) Nonparametric anova 1-way(Kruskal–Wallis) 2-way(Friedman) Ordered alternative(Jonckheere–Terpstra) Van der Waerden test

Bayesian inference

Correlation	Pearson product-moment Partial correlation Confounding variable Coefficient of determination
Regression analysis (see alsoTemplate:Least squares and regression analysis	Errors and residuals Regression validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS)
Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression
Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Homoscedasticity and Heteroscedasticity
Generalized linear model	Exponential families Logistic(Bernoulli) / Binomial / Poisson regressions
Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / multivariate / time-series / survival analysis

Categorical

Multivariate

Time-series

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality
Specific tests	Dickey–Fuller Johansen Q-statistic(Ljung–Box) Durbin–Watson Breusch–Godfrey
Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model(Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR) (Autoregressive model (AR))
Frequency domain	Spectral density estimation Fourier analysis Least-squares spectral analysis Wavelet Whittle likelihood

Survival

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time
Hazard function	Nelson–Aalen estimator
Test	Log-rank test

Applications

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics
Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification
Social statistics	Actuarial science Census Crime statistics Demography Econometrics Jurimetrics National accounts Official statistics Population statistics Psychometrics
Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging