s_data1 http://www.openmath.org/cd/s_data1.ocd 2003-04-01 2001-03-12 2 0 official relation1 fns2 arith1 set1 arith1 list1 set1 alg1 fns1 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 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 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 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 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 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 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