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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.
meanThis symbol represents an n-ary function denoting the mean of its arguments. That is, their sum divided by their number.
<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>
<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>
fns2.apply_to_list(s_data1.mean, $L) = fns2.apply_to_list(arith1.plus, $L) / set1.size($L)
apply_to_list ( mean , L ) = apply_to_list ( + , L ) size ( L )
<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> 3 </OMI>
</OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML"> <apply><csymbol cd="relation1">eq</csymbol> <apply><csymbol cd="s_data1">mean</csymbol> <cn type="integer">1</cn> <cn type="integer">2</cn> <cn type="integer">3</cn> </apply> <cn type="integer">3</cn> </apply> </math>
s_data1.mean(1, 2, 3) = 3
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.
<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>
<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 type="integer">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 type="integer">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>
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))
apply_to_list ( sdev ( L ) ) 2 = apply_to_list ( + , list ( ( x - mean ( L ) ) 2 | x ∈ L ) ) size ( )
<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>
<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>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 )
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)
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 )
This symbol represents a function requiring two or more arguments, denoting the variance of its arguments. That is, the square of the standard deviation.
<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>
<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 type="integer">2</cn>
</apply>
</apply>
</math>
fns2.apply_to_list(s_data1.variance($L)) = fns2.apply_to_list(s_data1.sdev($L)) ^ 2
apply_to_list ( variance ( L ) ) = apply_to_list ( sdev ( L ) ) 2
<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>
<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>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 )
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)
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 )
This symbol represents an n-ary function denoting the mode of its arguments. That is the value which occurs with the greatest frequency.
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML"> <apply><csymbol cd="relation1">eq</csymbol> <apply><csymbol cd="s_data1">mode</csymbol> <cn type="integer">1</cn> <cn type="integer">1</cn> <cn type="integer">2</cn> </apply> <cn type="integer">1</cn> </apply> </math>
s_data1.mode(1, 1, 2) = 1
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).
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML"> <apply><csymbol cd="relation1">eq</csymbol> <apply><csymbol cd="s_data1">median</csymbol> <cn type="integer">1</cn> <cn type="integer">2</cn> <cn type="integer">3</cn> </apply> <cn type="integer">2</cn> </apply> </math>
s_data1.median(1, 2, 3) = 2
median ( 1 , 2 , 3 ) = 2
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.
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML"> <apply><csymbol cd="s_data1">moment</csymbol> <cn type="integer">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>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 )
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)
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 )
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