This document is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. The copyright holder grants you permission to redistribute this document freely as a verbatim copy. Furthermore, the copyright holder permits you to develop any derived work from this document provided that the following conditions are met. a) The derived work acknowledges the fact that it is derived from this document, and maintains a prominent reference in the work to the original source. b) The fact that the derived work is not the original OpenMath document is stated prominently in the derived work. Moreover if both this document and the derived work are Content Dictionaries then the derived work must include a different CDName element, chosen so that it cannot be confused with any works adopted by the OpenMath Society. In particular, if there is a Content Dictionary Group whose name is, for example, `math' containing Content Dictionaries named `math1', `math2' etc., then you should not name a derived Content Dictionary `mathN' where N is an integer. However you are free to name it `private_mathN' or some such. This is because the names `mathN' may be used by the OpenMath Society for future extensions. c) The derived work is distributed under terms that allow the compilation of derived works, but keep paragraphs a) and b) intact. The simplest way to do this is to distribute the derived work under the OpenMath license, but this is not a requirement. If you have questions about this license please contact the OpenMath society at http://www.openmath.org. s_data1 http://www.openmath.org/cd http://www.openmath.org/cd/s_data1.ocd 2006-03-30 2004-03-30 3 0 official This CD holds the definitions of the basic statistical functions used on sample data. It is intended to be `compatible' with the MathML elements representing statistical functions, though it does not cover the concept of random variable which is mentioned in MathML. mean application This symbol represents an n-ary function denoting the mean of its arguments. That is, their sum divided by their number. The mean of n arguments is their sum divided by their number The mean of {1,2,3} is 3 1 2 3 3 sdev application This symbol represents a function requiring two or more arguments, denoting the sample standard deviation of its arguments. That is, the square root of (the sum of the squares of the deviations from the mean of the arguments, divided by the number of arguments). See CRC Standard Mathematical Tables and Formulae, editor: Dan Zwillinger, CRC Press Inc., 1996, (7.7.11) section 7.7.1. The square of the standard deviation of n arguments is the sum of the squares of the differences from their mean divided by the number of arguments. 2 2 This is an example to denote the standard deviation of a set of data variance application This symbol represents a function requiring two or more arguments, denoting the variance of its arguments. That is, the square of the standard deviation. The variance of n arguments is the square of the standard deviation of those arguments. 2 This is an example to denote the variance of a set of data mode application This symbol represents an n-ary function denoting the mode of its arguments. That is the value which occurs with the greatest frequency. The mode of n arguments is that value which occurs with the greatest frequency. The mode of {1,1,2} is 1 1 1 2 1 median application This symbol represents an n-ary function denoting the median of its arguments. That is, if the data were placed in ascending order then it denotes the middle one (in the case of an odd amount of data) or the average of the middle two (in the case of an even amount of data). The median of {1,2,3} is 2 1 2 3 2 moment application This symbol is used to denote the i'th moment of a set of data. The first argument should be the degree of the moment (that is, for the i'th moment the first argument should be i), the second argument should be the point about which the moment is being taken and the rest of the arguments are treated as the data. For n data values x_1, x_2, ..., x_n the i'th moment about c is (1/n) ((x_1-c)^i + (x_2-c)^i + ... + (x_n-c)^i). See CRC Standard Mathematical Tables and Formulae, editor: Dan Zwillinger, CRC Press Inc., 1996, section 7.7.1. This is an example to denote the 2'nd moment of a set of data about the origin. 2