词条 | Range (statistics) |
释义 |
In statistics, the range of a set of data is the difference between the largest and smallest values.[1] However, in descriptive statistics, this concept of range has a more complex meaning. The range is the size of the smallest interval (statistics) which contains all the data and provides an indication of statistical dispersion. It is measured in the same units as the data. Since it only depends on two of the observations, it is most useful in representing the dispersion of small data sets.[2] For continuous IID random variablesFor n independent and identically distributed continuous random variables X1, X2, ..., Xn 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). DistributionThe range has cumulative distribution function[3][4] 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."[3] If the distribution of each Xi 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 a Bessel function.[3] MomentsThe mean range is given by[5] where x(G) is the inverse function. In the case where each of the Xi has a standard normal distribution, the mean range is given by[6] For continuous non-IID random variablesFor n nonidentically distributed independent continuous random variables X1, X2, ..., Xn with cumulative distribution functions G1(x), G2(x), ..., Gn(x) and probability density functions g1(x), g2(x), ..., gn(x), the range has cumulative distribution function [4] For discrete IID random variablesFor n independent and identically distributed discrete random variables X1, X2, ..., Xn with cumulative distribution function G(x) and probability mass function g(x) the range of the Xi 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 Xi is {1,2,3,...,N} where N is a positive integer or infinity.[7][8] DistributionThe range has probability mass function[7][9][10] ExampleIf we suppose that g(x) = 1/N, the discrete uniform distribution for all x, then we find[9][11] DerivationThe probability of having a specific range value, t, can be determined by adding the probabilities of having two samples differing by t, and every other sample having a value between the two extremes. The probability of one sample having a value of x is . The probability of another having a value t greater than x is: . The probability of all other values lying between these two extremes is: . Combining the three together yields: Related quantitiesThe range is a simple function of the sample maximum and minimum and these are specific examples of order statistics. In particular, the range is a linear function of order statistics, which brings it into the scope of L-estimation. See also{{Portal|Statistics}}
References1. ^{{cite book|title=An Introduction to Statistics|author=George Woodbury|page=74|isbn=0534377556|publisher=Cengage Learning|year=2001}} {{Statistics|descriptive}}{{DEFAULTSORT:Range (Statistics)}}2. ^{{cite book|title=Elementary Statistics: Vol 2| pages=7–27 | author = Carin Viljoen| publisher=Pearson South Africa| year = 2000| isbn = 186891075X}} 3. ^1 2 {{cite journal | author = E. J. Gumbel | authorlink = E. J. Gumbel | year = 1947 | title = The Distribution of the Range | journal = The Annals of Mathematical Statistics | volume = 18 | issue = 3 | pages = 384–412 | jstor = 2235736 | doi=10.1214/aoms/1177730387}} 4. ^1 {{Cite book | last1 = Tsimashenka | first1 = I. | last2 = Knottenbelt | first2 = W. | last3 = Harrison | first3 = P. | authorlink3 = Peter G. Harrison| doi = 10.1007/978-3-642-30782-9_12 | chapter = Controlling Variability in Split-Merge Systems | title = Analytical and Stochastic Modeling Techniques and Applications | series = Lecture Notes in Computer Science | volume = 7314 | pages = 165 | year = 2012 | isbn = 978-3-642-30781-2 | pmid = | pmc = | url = http://www.doc.ic.ac.uk/~wjk/publications/tsimashenka-knottenbelt-harrison-asmta-2012.pdf}} 5. ^{{cite journal | author1 = H. O. Hartley | authorlink1 = H. O. Hartley | author2 = H. A. David | year = 1954 | title = Universal Bounds for Mean Range and Extreme Observation | journal = The Annals of Mathematical Statistics | volume = 25 | issue = 1 | pages = 85–99 | jstor = 2236514 | doi=10.1214/aoms/1177728848}} 6. ^{{cite journal | author = L. H. C. Tippett | authorlink = L. H. C. Tippett | year = 1925 | title = On the Extreme Individuals and the Range of Samples Taken from a Normal Population | journal = Biometrika | volume = 17 | issue = 3/4 | pages = 364–387 | jstor = 2332087 | doi=10.1093/biomet/17.3-4.364}} 7. ^1 {{Cite journal | last1 = Evans | first1 = D. L. | last2 = Leemis | first2 = L. M. | last3 = Drew | first3 = J. H. | title = The Distribution of Order Statistics for Discrete Random Variables with Applications to Bootstrapping | doi = 10.1287/ijoc.1040.0105 | journal = INFORMS Journal on Computing | volume = 18 | pages = 19 | year = 2006 | pmid = | pmc = }} 8. ^{{cite journal | author = Irving W. Burr | year = 1955 | title = Calculation of Exact Sampling Distribution of Ranges from a Discrete Population | journal = The Annals of Mathematical Statistics | volume = 26 | issue = 3 | pages = 530–532 | jstor = 2236482 | doi=10.1214/aoms/1177728500}} 9. ^1 {{Cite journal | last1 = Abdel-Aty | first1 = S. H. | title = Ordered variables in discontinuous distributions | doi = 10.1111/j.1467-9574.1954.tb00442.x | journal = Statistica Neerlandica | volume = 8 | issue = 2 | pages = 61–82 | year = 1954 | pmid = | pmc = }} 10. ^{{Cite journal | last1 = Siotani | first1 = M. | doi = 10.1007/BF02863574 | title = Order statistics for discrete case with a numerical application to the binomial distribution | journal = Annals of the Institute of Statistical Mathematics | volume = 8 | pages = 95–96 | year = 1956 | pmid = | pmc = }} 11. ^{{cite journal | author = Paul R. Rider | year = 1951 | title = The Distribution of the Range in Samples from a Discrete Rectangular Population | journal = Journal of the American Statistical Association | volume = 46 | issue = 255 | pages = 375–378 | jstor = 2280515 | doi=10.1080/01621459.1951.10500796}} 3 : Statistical deviation and dispersion|Scale statistics|Summary statistics |
随便看 |
|
开放百科全书收录14589846条英语、德语、日语等多语种百科知识,基本涵盖了大多数领域的百科知识,是一部内容自由、开放的电子版国际百科全书。