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# OpenMath Content Dictionary: s_data1

Canonical URL:
http://www.openmath.org/cd/s_data1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
s_data1.ocd
CD as XML Encoded OpenMath:
s_data1.omcd
Defines:
mean, median, mode, moment, sdev, variance
Date:
2004-03-30
Version:
3
Review Date:
2006-03-30
Status:
official

```
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.
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.
society at http://www.openmath.org.
```

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

Role:
application
Description:

This symbol represents an n-ary function denoting the mean of its arguments. That is, their sum divided by their number.

Commented Mathematical property (CMP):
The mean of n arguments is their sum divided by their number
Formal Mathematical property (FMP):
$\mathrm{apply_to_list}\left(\mathrm{mean},L\right)=\frac{\mathrm{apply_to_list}\left(+,L\right)}{\mathrm{size}\left(L\right)}$
Example:
The mean of {1,2,3} is 2
$\mathrm{mean}\left(1,2,3\right)=2$
Signatures:
sts

 [Next: sdev] [Last: moment] [Top]

## sdev

Role:
application
Description:

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.

Commented Mathematical property (CMP):
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.
Formal Mathematical property (FMP):
${\mathrm{apply_to_list}\left(\mathrm{sdev}\left(L\right)\right)}^{2}=\frac{\mathrm{apply_to_list}\left(+,\mathrm{list}\left({\left(x-\mathrm{mean}\left(L\right)\right)}^{2}|x\in L\right)\right)}{\mathrm{size}\left(\right)}$
Example:
This is an example to denote the standard deviation of a set of data
$\mathrm{sdev}\left(3.1,2.2,1.8,1.1,3.3,2.4,5.5,2.3,1.7,1.8,3.4,4.0,3.3\right)$
Signatures:
sts

 [Next: variance] [Previous: mean] [Top]

## variance

Role:
application
Description:

This symbol represents a function requiring two or more arguments, denoting the variance of its arguments. That is, the square of the standard deviation.

Commented Mathematical property (CMP):
The variance of n arguments is the square of the standard deviation of those arguments.
Formal Mathematical property (FMP):
$\mathrm{apply_to_list}\left(\mathrm{variance}\left(L\right)\right)={\mathrm{apply_to_list}\left(\mathrm{sdev}\left(L\right)\right)}^{2}$
Example:
This is an example to denote the variance of a set of data
$\mathrm{variance}\left(3.1,2.2,1.8,1.1,3.3,2.4,5.5,2.3,1.7,1.8,3.4,4.0,3.3\right)$
Signatures:
sts

 [Next: mode] [Previous: sdev] [Top]

## mode

Role:
application
Description:

This symbol represents an n-ary function denoting the mode of its arguments. That is the value which occurs with the greatest frequency.

Commented Mathematical property (CMP):
The mode of n arguments is that value which occurs with the greatest frequency.
Example:
The mode of {1,1,2} is 1
$\mathrm{mode}\left(1,1,2\right)=1$
Signatures:
sts

 [Next: median] [Previous: variance] [Top]

## median

Role:
application
Description:

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).

Example:
The median of {1,2,3} is 2
$\mathrm{median}\left(1,2,3\right)=2$
Signatures:
sts

 [Next: moment] [Previous: mode] [Top]

## moment

Role:
application
Description:

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.

Example:
This is an example to denote the 2'nd moment of a set of data about the origin.
$\mathrm{moment}\left(2,0,3.1,2.2,1.8,1.1,3.3,2.4,5.5,2.3,1.7,1.8,3.4,4.0,3.3\right)$
Signatures:
sts

 [First: mean] [Previous: median] [Top]

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