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.
       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.

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):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="fns2" name="apply_to_list"/>
      <OMS cd="s_data1" name="mean"/>
      <OMV name="L"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="divide"/>
      <OMA>
        <OMS cd="fns2" name="apply_to_list"/>
	<OMS cd="arith1" name="plus"/>
	<OMV name="L"/>
      </OMA>
      <OMA>
        <OMS cd="set1" name="size"/>
	<OMV name="L"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="fns2">apply_to_list</csymbol><csymbol cd="s_data1">mean</csymbol><ci>L</ci></apply>
  <apply><csymbol cd="arith1">divide</csymbol>
   <apply><csymbol cd="fns2">apply_to_list</csymbol><csymbol cd="arith1">plus</csymbol><ci>L</ci></apply>
   <apply><csymbol cd="set1">size</csymbol><ci>L</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (apply_to_list (mean, L) , divide (apply_to_list (plus, L) , size ( L) ) )

Popcorn

fns2.apply_to_list(s_data1.mean, $L) = fns2.apply_to_list(arith1.plus, $L) / set1.size($L)

Rendered Presentation MathML

$$\mathrm{apply\_to\_list}(\mathrm{mean},L)=\frac{\mathrm{apply\_to\_list}(+,L)}{\mathrm{size}\left(L\right)}$$

Example:

The mean of {1,2,3} is 2

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="s_data1" name="mean"/>
      <OMI> 1 </OMI> <OMI> 2 </OMI> <OMI> 3 </OMI>
    </OMA>
    <OMI> 2 </OMI>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="s_data1">mean</csymbol><cn>1</cn><cn>2</cn><cn>3</cn></apply>
  <cn>2</cn>
 </apply>
</math>

Prefix

eq (mean ( 1 , 2 , 3 ) , 2 )

Popcorn

s_data1.mean(1, 2, 3) = 2

Rendered Presentation MathML

$$\mathrm{mean}(1,2,3)=2$$

Signatures:

sts

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):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="arith1" name="power"/>
      <OMA>
        <OMS cd="fns2" name="apply_to_list"/>
	<OMA>
	  <OMS cd="s_data1" name="sdev"/>
	  <OMV name="L"/>
	</OMA>
      </OMA>
      <OMI> 2 </OMI>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="divide"/>
      <OMA>
        <OMS cd="fns2" name="apply_to_list"/>
	<OMS cd="arith1" name="plus"/>
	<OMA>
	  <OMS cd="list1" name="map"/>
	  <OMBIND>
	    <OMS cd="fns1" name="lambda"/>
	    <OMBVAR>
	      <OMV name="x"/>
	    </OMBVAR>
	    <OMA>
	      <OMS cd="arith1" name="power"/>
	      <OMA>
	        <OMS cd="arith1" name="minus"/>
		<OMV name="x"/>
		<OMA>
		  <OMS cd="s_data1" name="mean"/>
		  <OMV name="L"/>
		</OMA>
	      </OMA>
	      <OMI> 2 </OMI>
	    </OMA>
	  </OMBIND>
	  <OMV name="L"/>
	</OMA>
      </OMA>
      <OMA>
        <OMS cd="set1" name="size"/>
	<OMA>
	  <OMS cd="fns2" name="apply_to_list"/>
	  <OMS cd="set1" name="set"/>
	  <OMV name="L"/>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="arith1">power</csymbol>
   <apply><csymbol cd="fns2">apply_to_list</csymbol>
    <apply><csymbol cd="s_data1">sdev</csymbol><ci>L</ci></apply>
   </apply>
   <cn>2</cn>
  </apply>
  <apply><csymbol cd="arith1">divide</csymbol>
   <apply><csymbol cd="fns2">apply_to_list</csymbol>
    <csymbol cd="arith1">plus</csymbol>
    <apply><csymbol cd="list1">map</csymbol>
     <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>x</ci></bvar>
      <apply><csymbol cd="arith1">power</csymbol>
       <apply><csymbol cd="arith1">minus</csymbol>
        <ci>x</ci>
        <apply><csymbol cd="s_data1">mean</csymbol><ci>L</ci></apply>
       </apply>
       <cn>2</cn>
      </apply>
     </bind>
     <ci>L</ci>
    </apply>
   </apply>
   <apply><csymbol cd="set1">size</csymbol>
    <apply><csymbol cd="fns2">apply_to_list</csymbol><csymbol cd="set1">set</csymbol><ci>L</ci></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (power (apply_to_list (sdev ( L) ) , 2 ) , divide (apply_to_list (plus, map (lambda [ x ] . (power (minus ( x, mean ( L) ) , 2 ) ) , L) ) , size (apply_to_list (set, L) ) ) )

Popcorn

fns2.apply_to_list(s_data1.sdev($L)) ^ 2 = fns2.apply_to_list(arith1.plus, list1.map(fns1.lambda[$x -> ($x - s_data1.mean($L)) ^ 2], $L)) / set1.size(fns2.apply_to_list(set1.set, $L))

Rendered Presentation MathML

$${\mathrm{apply\_to\_list}\left(\mathrm{sdev}\left(L\right)\right)}^2=\frac{\mathrm{apply\_to\_list}(+,\mathrm{list}\left({(x-\mathrm{mean}\left(L\right))}^2\right|x\in L))}{\mathrm{size}\left(\right)}$$

Example:

This is an example to denote the standard deviation of a set of data

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="s_data1" name="sdev"/>
    <OMF dec="3.1"/> <OMF dec="2.2"/> <OMF dec="1.8"/> <OMF dec="1.1"/>
    <OMF dec="3.3"/> <OMF dec="2.4"/> <OMF dec="5.5"/> <OMF dec="2.3"/>
    <OMF dec="1.7"/> <OMF dec="1.8"/> <OMF dec="3.4"/> <OMF dec="4.0"/>
    <OMF dec="3.3"/>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="s_data1">sdev</csymbol>
  <cn type="real">3.1</cn>
  <cn type="real">2.2</cn>
  <cn type="real">1.8</cn>
  <cn type="real">1.1</cn>
  <cn type="real">3.3</cn>
  <cn type="real">2.4</cn>
  <cn type="real">5.5</cn>
  <cn type="real">2.3</cn>
  <cn type="real">1.7</cn>
  <cn type="real">1.8</cn>
  <cn type="real">3.4</cn>
  <cn type="real">4.0</cn>
  <cn type="real">3.3</cn>
 </apply>
</math>

Prefix

sdev ( 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 )

Popcorn

s_data1.sdev(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)

Rendered Presentation MathML

$$\mathrm{sdev}(3.1,2.2,1.8,1.1,3.3,2.4,5.5,2.3,1.7,1.8,3.4,4.0,3.3)$$

Signatures:

sts

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):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="fns2" name="apply_to_list"/>
      <OMA>
        <OMS cd="s_data1" name="variance"/>
	<OMV name="L"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="power"/>
      <OMA>
        <OMS cd="fns2" name="apply_to_list"/>
	<OMA>
	  <OMS cd="s_data1" name="sdev"/>
	  <OMV name="L"/>
	</OMA>
      </OMA>
      <OMI> 2 </OMI>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="fns2">apply_to_list</csymbol>
   <apply><csymbol cd="s_data1">variance</csymbol><ci>L</ci></apply>
  </apply>
  <apply><csymbol cd="arith1">power</csymbol>
   <apply><csymbol cd="fns2">apply_to_list</csymbol>
    <apply><csymbol cd="s_data1">sdev</csymbol><ci>L</ci></apply>
   </apply>
   <cn>2</cn>
  </apply>
 </apply>
</math>

Prefix

eq (apply_to_list (variance ( L) ) , power (apply_to_list (sdev ( L) ) , 2 ) )

Popcorn

fns2.apply_to_list(s_data1.variance($L)) = fns2.apply_to_list(s_data1.sdev($L)) ^ 2

Rendered Presentation MathML

$$\mathrm{apply\_to\_list}\left(\mathrm{variance}\left(L\right)\right)={\mathrm{apply\_to\_list}\left(\mathrm{sdev}\left(L\right)\right)}^2$$

Example:

This is an example to denote the variance of a set of data

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="s_data1" name="variance"/>
    <OMF dec="3.1"/> <OMF dec="2.2"/> <OMF dec="1.8"/> <OMF dec="1.1"/>
    <OMF dec="3.3"/> <OMF dec="2.4"/> <OMF dec="5.5"/> <OMF dec="2.3"/>
    <OMF dec="1.7"/> <OMF dec="1.8"/> <OMF dec="3.4"/> <OMF dec="4.0"/>
    <OMF dec="3.3"/>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="s_data1">variance</csymbol>
  <cn type="real">3.1</cn>
  <cn type="real">2.2</cn>
  <cn type="real">1.8</cn>
  <cn type="real">1.1</cn>
  <cn type="real">3.3</cn>
  <cn type="real">2.4</cn>
  <cn type="real">5.5</cn>
  <cn type="real">2.3</cn>
  <cn type="real">1.7</cn>
  <cn type="real">1.8</cn>
  <cn type="real">3.4</cn>
  <cn type="real">4.0</cn>
  <cn type="real">3.3</cn>
 </apply>
</math>

Prefix

variance ( 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 )

Popcorn

s_data1.variance(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)

Rendered Presentation MathML

$$\mathrm{variance}(3.1,2.2,1.8,1.1,3.3,2.4,5.5,2.3,1.7,1.8,3.4,4.0,3.3)$$

Signatures:

sts

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

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="s_data1" name="mode"/>
      <OMI> 1 </OMI> <OMI> 1 </OMI> <OMI> 2 </OMI>
    </OMA>
    <OMI> 1 </OMI>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="s_data1">mode</csymbol><cn>1</cn><cn>1</cn><cn>2</cn></apply>
  <cn>1</cn>
 </apply>
</math>

Prefix

eq (mode ( 1 , 1 , 2 ) , 1 )

Popcorn

s_data1.mode(1, 1, 2) = 1

Rendered Presentation MathML

$$\mathrm{mode}(1,1,2)=1$$

Signatures:

sts

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

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="s_data1" name="median"/>
      <OMI> 1 </OMI> <OMI> 2 </OMI> <OMI> 3 </OMI>
    </OMA>
    <OMI> 2 </OMI>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="s_data1">median</csymbol><cn>1</cn><cn>2</cn><cn>3</cn></apply>
  <cn>2</cn>
 </apply>
</math>

Prefix

eq (median ( 1 , 2 , 3 ) , 2 )

Popcorn

s_data1.median(1, 2, 3) = 2

Rendered Presentation MathML

$$\mathrm{median}(1,2,3)=2$$

Signatures:

sts

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.

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="s_data1" name="moment"/>
    <OMI> 2 </OMI>
    <OMS cd="alg1" name="zero"/>
    <OMF dec="3.1"/> <OMF dec="2.2"/> <OMF dec="1.8"/> <OMF dec="1.1"/>
    <OMF dec="3.3"/> <OMF dec="2.4"/> <OMF dec="5.5"/> <OMF dec="2.3"/>
    <OMF dec="1.7"/> <OMF dec="1.8"/> <OMF dec="3.4"/> <OMF dec="4.0"/>
    <OMF dec="3.3"/>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="s_data1">moment</csymbol>
  <cn>2</cn>
  <csymbol cd="alg1">zero</csymbol>
  <cn type="real">3.1</cn>
  <cn type="real">2.2</cn>
  <cn type="real">1.8</cn>
  <cn type="real">1.1</cn>
  <cn type="real">3.3</cn>
  <cn type="real">2.4</cn>
  <cn type="real">5.5</cn>
  <cn type="real">2.3</cn>
  <cn type="real">1.7</cn>
  <cn type="real">1.8</cn>
  <cn type="real">3.4</cn>
  <cn type="real">4.0</cn>
  <cn type="real">3.3</cn>
 </apply>
</math>

Prefix

moment ( 2 , zero, 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 )

Popcorn

s_data1.moment(2, alg1.zero, 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)

Rendered Presentation MathML

$$\mathrm{moment}(2,0,3.1,2.2,1.8,1.1,3.3,2.4,5.5,2.3,1.7,1.8,3.4,4.0,3.3)$$

Signatures:

sts