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width="327" height="120" align-x="36" align-y="150" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Outliers are data points that seem not to be genuine because they lie far outside of the expected behavior of the data. 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The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">trim</c></inlineAttr></b></f> function can be used to remove rows with specified indices from a data set.</p></text></region><region region-id="1229" left="36" top="272.25" width="317.25" height="24" align-x="36" align-y="282" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Suppose we wish to determine if the following data set has any outliers.</p></text></region><region region-id="1214" left="66" top="303" width="131.25" height="12.75" align-x="77.25" align-y="312" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>y</ml:id><ml:apply><ml:id>READPRN</ml:id><ml:str>heatflow1.txt</ml:str></ml:apply></ml:define></math><rendering item-idref="4" /></region><region region-id="853" left="222" top="303" width="63.75" height="12.75" align-x="240" align-y="312" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>N0</ml:id><ml:apply><ml:id>length</ml:id><ml:id>y</ml:id></ml:apply></ml:define></math><rendering item-idref="5" /></region><region region-id="856" left="222" top="321" width="43.5" height="12.75" align-x="239.25" align-y="330" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>N0</ml:id></ml:eval></math><rendering item-idref="6" /></region><region region-id="825" left="36" top="344.25" width="296.25" height="48" align-x="36" align-y="354" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">We can examine a scatter plot to spot potential outliers, but unless the outliers are severe and infrequent, we may not see them. It is also not a quantitative metric for saying specifically which points are outliers.</p></text></region><region region-id="1336" left="66" top="405" width="55.5" height="12.75" align-x="74.25" align-y="414" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>i</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:minus /><ml:id>N0</ml:id><ml:real>1</ml:real></ml:apply></ml:range></ml:define></math><rendering item-idref="7" /></region><region region-id="1337" left="222" top="405" width="85.5" height="12.75" align-x="261" align-y="414" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>mean</ml:id><ml:id>y</ml:id></ml:apply></ml:eval></math><rendering item-idref="8" /></region><region region-id="1338" left="222" top="423" width="87.75" height="12.75" align-x="263.25" align-y="432" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>Stdev</ml:id><ml:id>y</ml:id></ml:apply></ml:eval></math><rendering item-idref="9" /></region><region region-id="832" left="12" top="444" width="330.75" height="189.75" align-x="12" align-y="444" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="10" /><rendering item-idref="11" /></region><region region-id="1225" left="36" top="638.25" width="231" height="12" align-x="36" align-y="648" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The mean of the data is indicated on the plot.</p></text></region><region region-id="835" left="0" top="660" width="6000" height="6" align-x="0" align-y="660" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1145" left="36" top="667.5" width="106.5" height="12.75" align-x="47.25" align-y="678" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Grubb's outlier criteria</p></text><indexes><phrase>Grubb's outlier criteria</phrase></indexes></region><region region-id="729" left="36" top="692.25" width="320.25" height="48" align-x="39" align-y="702" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Grubb's outlier test answers the question, "If all values in a data set were sampled from a normally-distributed population, what is the chance of randomly obtaining any particular value so far from the other values?". The test statistic is given by </p></text></region><region region-id="1227" left="66" top="752.25" width="109.5" height="33.75" align-x="83.25" align-y="774" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Gb</ml:id><ml:apply><ml:div /><ml:apply><ml:id>max</ml:id><ml:apply><ml:vectorize /><ml:apply><ml:absval /><ml:apply><ml:minus /><ml:id>y</ml:id><ml:apply><ml:id>mean</ml:id><ml:id>y</ml:id></ml:apply></ml:apply></ml:apply></ml:apply></ml:apply><ml:apply><ml:id>stdev</ml:id><ml:id>y</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="12" /></region><region region-id="1228" left="228" top="765" width="63" height="12.75" align-x="244.5" align-y="774" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>Gb</ml:id></ml:eval></math><rendering item-idref="13" /></region><region region-id="1192" left="36" top="794.25" width="310.5" height="60" align-x="36" align-y="804" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">that is, what is the <f family="Verdana" charset="0"><i><inlineAttr line-through="false"><c val="#000">maximum</c></inlineAttr></i></f><f family="Times New Roman" charset="0"><i><inlineAttr line-through="false"><c val="#000"><sp /></c></inlineAttr></i></f>deviation of any single data point from the mean of the dataset, divided by the <f family="Verdana" charset="0"><i><inlineAttr line-through="false"><c val="#000">standard</c></inlineAttr></i></f> deviation from the mean. This statistic is compared to the probability that this maximum value lies outside of some percentage of the distribution. The test criterion is</p></text></region><region region-id="1193" left="66" top="860.25" width="36" height="13.5" align-x="78.75" align-y="870" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>α</ml:id><ml:real>0.02</ml:real></ml:define></math><rendering item-idref="14" /></region><region region-id="1194" left="114" top="860.25" width="221.25" height="12" align-x="114" align-y="870" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">98% confidence of outlier test being correct</p></text></region><region region-id="838" left="66" top="885" width="222.75" height="72.75" align-x="111" align-y="924" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:function><ml:id>crit</ml:id><ml:boundVars><ml:id>α</ml:id><ml:id>N</ml:id></ml:boundVars></ml:function><ml:apply><ml:mult style="default" /><ml:apply><ml:div /><ml:apply><ml:minus /><ml:id>N0</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:sqrt /><ml:id>N0</ml:id></ml:apply></ml:apply><ml:parens><ml:apply><ml:sqrt /><ml:apply><ml:div /><ml:apply><ml:pow /><ml:apply><ml:id>qt</ml:id><ml:apply><ml:div /><ml:id>α</ml:id><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:id>N0</ml:id></ml:apply></ml:apply><ml:apply><ml:minus /><ml:id>N0</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:real>2</ml:real></ml:apply><ml:apply><ml:plus /><ml:apply><ml:minus /><ml:id>N</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:pow /><ml:apply><ml:id>qt</ml:id><ml:apply><ml:div /><ml:id>α</ml:id><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:id>N0</ml:id></ml:apply></ml:apply><ml:apply><ml:minus /><ml:id>N0</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:apply></ml:parens></ml:apply></ml:define></math><rendering item-idref="15" /></region><region region-id="1355" left="294" top="914.25" width="95.25" height="13.5" align-x="342.75" align-y="924" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>crit</ml:id><ml:id>α</ml:id><ml:id>N0</ml:id></ml:apply></ml:eval></math><rendering item-idref="16" /></region><region region-id="1356" left="294" top="938.25" width="88.5" height="13.5" align-x="365.25" align-y="948" show-border="false" show-highlight="true" is-protected="true" background-color="#80ffff" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:greaterThan /><ml:id>Gb</ml:id><ml:apply><ml:id>crit</ml:id><ml:id>α</ml:id><ml:id>N0</ml:id></ml:apply></ml:apply></ml:eval></math><rendering item-idref="17" /></region><region region-id="1146" left="36" top="968.25" width="306" height="36" align-x="36" align-y="978" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">where <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">qt </c></inlineAttr></b></f>is the inverse cumulative probability density of the student's t function. The test finds no outliers are present if we wish to be 98% confident of the analysis.</p></text></region><region region-id="1147" left="66" top="1016.25" width="121.5" height="13.5" align-x="78.75" align-y="1026" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id>α</ml:id><ml:apply><ml:id>root</ml:id><ml:apply><ml:minus /><ml:apply><ml:id>crit</ml:id><ml:id>α</ml:id><ml:id>N0</ml:id></ml:apply><ml:id>Gb</ml:id></ml:apply><ml:id>α</ml:id></ml:apply></ml:define></math><rendering item-idref="18" /></region><region region-id="1148" left="192" top="1016.25" width="54" height="13.5" align-x="204" align-y="1026" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>α</ml:id></ml:eval></math><rendering item-idref="19" /></region><region region-id="1149" left="258" top="1016.25" width="131.25" height="50.25" align-x="258" align-y="1026" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">outliers will be detected when we only wish to be <region region-id="1329" left="258" top="1040.25" width="427.5" height="13.5" align-x="285.75" align-y="1050" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:minus /><ml:real>1</ml:real><ml:id>α</ml:id></ml:apply><ml:unitOverride><ml:id>%</ml:id></ml:unitOverride></ml:eval><resultFormat><general radix="dec" complex-threshold="10" precision="2" show-trailing-zeros="false" exponential-threshold="3" /><matrix display-style="auto" expand-nested-arrays="false" /><unit format-units="true" simplify-units="true" /></resultFormat></math><rendering item-idref="20" /></region> sure of our test.</p></text></region><region region-id="1150" left="66" top="1040.25" width="120.75" height="13.5" align-x="169.5" align-y="1050" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:greaterThan /><ml:id>Gb</ml:id><ml:apply><ml:id>crit</ml:id><ml:apply><ml:plus /><ml:id>α</ml:id><ml:real>.0001</ml:real></ml:apply><ml:id>N0</ml:id></ml:apply></ml:apply></ml:eval></math><rendering item-idref="21" /></region><region region-id="1151" left="36" top="1070.25" width="288" height="24" align-x="36" align-y="1080" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This is designed to show you what the test does, but the functionality is wrapped up in the <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">Grubbs</c></inlineAttr></b></f> function: </p></text></region><region region-id="1221" left="66" top="1113" width="101.25" height="12.75" align-x="97.5" align-y="1122" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>outlier</ml:id><ml:apply><ml:id>Grubbs</ml:id><ml:id>y</ml:id><ml:real>0.90</ml:real></ml:apply></ml:define></math><rendering item-idref="22" /></region><region region-id="1222" left="186" top="1104.75" width="153" height="29.25" align-x="216.75" align-y="1122" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>outlier</ml:id></ml:eval></math><rendering item-idref="23" /></region><region region-id="1133" left="36" top="1142.25" width="308.25" height="108" align-x="36" align-y="1152" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The first column of output gives the index(ices) of the points identified as outliers, those whose test statistic (distance from the mean divided by standard deviation) exceeded the Grubb's test criterion. The second column gives the value of the test statistic at those points, and the third column gives the distance of each data point's test statistic from critical Grubbs' value. If no test point exceeds the criteria, the function returns the data point which is closest to meeting the criterion.</p></text></region><region region-id="1340" left="66" top="1257" width="78" height="19.5" align-x="111" align-y="1266" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:indexer /><ml:id>y</ml:id><ml:apply><ml:indexer /><ml:id>outlier</ml:id><ml:sequence><ml:real>0</ml:real><ml:real>0</ml:real></ml:sequence></ml:apply></ml:apply></ml:eval></math><rendering item-idref="24" /></region><region region-id="1341" left="66" top="1281" width="141.75" height="18" align-x="161.25" align-y="1290" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:plus /><ml:apply><ml:indexer /><ml:id>outlier</ml:id><ml:sequence><ml:real>0</ml:real><ml:real>1</ml:real></ml:sequence></ml:apply><ml:apply><ml:indexer /><ml:id>outlier</ml:id><ml:sequence><ml:real>0</ml:real><ml:real>2</ml:real></ml:sequence></ml:apply></ml:apply></ml:eval></math><rendering item-idref="25" /></region><region region-id="1339" left="228" top="1281" width="95.25" height="12.75" align-x="276.75" align-y="1290" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>crit</ml:id><ml:real>.1</ml:real><ml:id>N0</ml:id></ml:apply></ml:eval></math><rendering item-idref="26" /></region><region region-id="1342" left="66" top="1305" width="141" height="18" align-x="169.5" align-y="1314" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>outlier</ml:id><ml:sequence><ml:real>0</ml:real><ml:real>1</ml:real></ml:sequence></ml:apply><ml:apply><ml:id>crit</ml:id><ml:real>0.1</ml:real><ml:id>N0</ml:id></ml:apply></ml:apply></ml:eval></math><rendering item-idref="27" /></region><region region-id="1333" left="0" top="1326" width="6000" height="6" align-x="0" align-y="1326" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1209" left="36" top="1340.25" width="297.75" height="96" align-x="36" align-y="1350" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Even if more than one index is returned, this does not mean that all candidates must be outliers, since the critical value, G, and test statistic change if a candidate is removed: both are dependent on N. Try changing the probability that data randomly takes given value in the Grubbs call above to 0.95, and 0.85, to see which points are returned. It is interesting to try removing some of those points from the data, and rerunning the test.</p></text></region><region region-id="1018" left="36" top="1442.25" width="297" height="24" align-x="36" align-y="1452" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If you just want the single point most likely to be an outlier, use</p></text></region><region region-id="1019" left="66" top="1479" width="213.75" height="12.75" align-x="162" align-y="1488" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>GrubbsClassic</ml:id><ml:id>y</ml:id><ml:real>0.90</ml:real></ml:apply></ml:eval></math><rendering item-idref="28" /></region><region region-id="1353" left="36" top="1496.25" width="306" height="36" align-x="36" align-y="1506" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that these functions can accept a matrix of input, in which case they will return nested pairs of indices for the array locations of outliers.</p></text></region><region region-id="1354" left="36" top="1544.25" width="303" height="48" align-x="39" align-y="1554" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Because this test assumes normality of the data, it's a good idea to check your data, perhaps using a visual test like the <b><u><link href="./QQplot.xmcd#probplot" title="Links to section" popup="false">Normal Probability plot</link></u></b>, before proceeding. The test is not truly valid on non-normally distributed data.</p></text></region><region region-id="1159" left="36" top="1597.5" width="80.25" height="12.75" align-x="36" align-y="1608" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Three Sigma rule</p></text></region><region region-id="1160" left="36" top="1615.5" width="309" height="48.75" align-x="36" align-y="1626" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The 3<f family="Symbol" charset="2" size="11"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">s</c></inlineAttr></f> test is much the same as Grubbs', but checks to see if the distance from the mean divided by the standard deviation for each data point is greater than 3, rather than greater than the Grubb's statistic.</p></text></region><region region-id="1204" left="66" top="1672.5" width="146.25" height="45.75" align-x="131.25" align-y="1698" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>ThreeSigma</ml:id><ml:id>y</ml:id></ml:apply></ml:eval></math><rendering item-idref="29" /></region><region region-id="1163" left="36" top="1724.25" width="315.75" height="36" align-x="36" align-y="1734" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If no outliers are detected, the point closest to being an outlier is returned, with its index in the first column, and its test statistic in the second column.</p></text></region><region region-id="1166" left="66" top="1773" width="114" height="33" align-x="133.5" align-y="1794" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:div /><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>y</ml:id><ml:real>19</ml:real></ml:apply><ml:apply><ml:id>mean</ml:id><ml:id>y</ml:id></ml:apply></ml:apply><ml:apply><ml:id>stdev</ml:id><ml:id>y</ml:id></ml:apply></ml:apply></ml:eval></math><rendering item-idref="30" /></region><region region-id="878" left="36" top="1819.5" width="147" height="12.75" align-x="36" align-y="1830" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Interquartile range and outliers</p></text><indexes><phrase>interquartile range and outliers</phrase><phrase>outliers, interquartile range and</phrase></indexes></region><region region-id="1230" left="36" top="1844.25" width="310.5" height="24" align-x="36" align-y="1854" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">A different definition of outliers, based on percentiles, can also be used:</p></text></region><region region-id="886" left="66" top="1887" width="108" height="12.75" align-x="94.5" align-y="1896" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:function><ml:id>Q1</ml:id><ml:boundVars><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>percentile</ml:id><ml:id>x</ml:id><ml:apply><ml:mult style="auto-select" /><ml:real>25</ml:real><ml:id>%</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="31" /></region><region region-id="887" left="228" top="1887" width="108" height="12.75" align-x="256.5" align-y="1896" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:function><ml:id>Q3</ml:id><ml:boundVars><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>percentile</ml:id><ml:id>x</ml:id><ml:apply><ml:mult style="auto-select" /><ml:real>75</ml:real><ml:id>%</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="32" /></region><region region-id="883" left="36" top="1916.25" width="327" height="36" align-x="36" align-y="1926" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Box plots will label points as extreme in range if they lie 1.5 times the interquartile range above or below the end points of that range.</p></text></region><region region-id="1232" left="0" top="1962" width="6000" height="6" align-x="0" align-y="1962" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="884" left="66" top="1977" width="268.5" height="155.25" align-x="113.25" align-y="1986" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:function><ml:id>extreme</ml:id><ml:boundVars><ml:id>x</ml:id></ml:boundVars></ml:function><ml:program><ml:localDefine><ml:id>IQ</ml:id><ml:apply><ml:minus /><ml:apply><ml:id>Q3</ml:id><ml:id>x</ml:id></ml:apply><ml:apply><ml:id>Q1</ml:id><ml:id>x</ml:id></ml:apply></ml:apply></ml:localDefine><ml:localDefine><ml:id>lower_whisker</ml:id><ml:apply><ml:minus /><ml:apply><ml:id>Q1</ml:id><ml:id>x</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:real>1.5</ml:real><ml:id>IQ</ml:id></ml:apply></ml:apply></ml:localDefine><ml:localDefine><ml:id>upper_whisker</ml:id><ml:apply><ml:plus /><ml:apply><ml:id>Q3</ml:id><ml:id>x</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:real>1.5</ml:real><ml:id>IQ</ml:id></ml:apply></ml:apply></ml:localDefine><ml:localDefine><ml:id>i</ml:id><ml:real>0</ml:real></ml:localDefine><ml:for><ml:id>j</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:id>last</ml:id><ml:id>x</ml:id></ml:apply></ml:range><ml:ifThen><ml:apply><ml:plus /><ml:parens><ml:apply><ml:lessThan /><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>j</ml:id></ml:apply><ml:id>lower_whisker</ml:id></ml:apply></ml:parens><ml:parens><ml:apply><ml:greaterThan /><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>j</ml:id></ml:apply><ml:id>upper_whisker</ml:id></ml:apply></ml:parens></ml:apply><ml:program><ml:localDefine><ml:apply><ml:indexer /><ml:id>o</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>j</ml:id></ml:apply></ml:localDefine><ml:localDefine><ml:id>i</ml:id><ml:apply><ml:plus /><ml:id>i</ml:id><ml:real>1</ml:real></ml:apply></ml:localDefine></ml:program></ml:ifThen></ml:for><ml:id>o</ml:id></ml:program></ml:define></math><rendering item-idref="33" /></region><region region-id="891" left="36" top="2144.25" width="252.75" height="12" align-x="36" align-y="2154" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This method detects four outliers in the test data:</p></text></region><region region-id="902" left="66" top="2163" width="86.25" height="12.75" align-x="102" align-y="2172" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>outliers</ml:id><ml:apply><ml:id>extreme</ml:id><ml:id>y</ml:id></ml:apply></ml:define></math><rendering item-idref="34" /></region><region region-id="903" left="192" top="2163" width="107.25" height="12.75" align-x="201" align-y="2172" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>j</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:minus /><ml:apply><ml:id>length</ml:id><ml:id>outliers</ml:id></ml:apply><ml:real>1</ml:real></ml:apply></ml:range></ml:define></math><rendering item-idref="35" /></region><region region-id="904" left="66" top="2187" width="116.25" height="18" align-x="96" align-y="2196" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>index</ml:id><ml:id>j</ml:id></ml:apply><ml:apply><ml:indexer /><ml:apply><ml:id>match</ml:id><ml:apply><ml:indexer /><ml:id>outliers</ml:id><ml:id>j</ml:id></ml:apply><ml:id>y</ml:id></ml:apply><ml:real>0</ml:real></ml:apply></ml:define></math><rendering item-idref="36" /></region><region region-id="919" left="144" top="2216.25" width="186" height="36" align-x="144" align-y="2226" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">These are indicated with blue boxes in the scatterplot below, as are the limits of the interquartile range.</p></text></region><region region-id="907" left="66" top="2210.25" width="66.75" height="62.25" align-x="92.25" align-y="2244" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>index</ml:id></ml:eval></math><rendering item-idref="37" /></region><region region-id="1174" left="48" top="2280" width="321.75" height="189.75" align-x="48" align-y="2280" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="38" /><rendering item-idref="39" /></region><region region-id="1323" left="36" top="2474.25" width="315.75" height="36" align-x="36" align-y="2484" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">There are also methods for detecting outliers after data has been fit to a function, <b><u><link href="./residuals.xmcd#outlier" title="Links to section" popup="false">using the residuals</link></u></b>, which are worth noting as an alternative method to those shown above.</p></text></region><region region-id="1178" left="0" top="2526" width="6000" height="6" align-x="0" align-y="2526" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1180" left="36" top="2545.5" width="78" height="12.75" align-x="36" align-y="2556" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Trimming values</p></text><indexes><phrase>trimming values</phrase></indexes></region><region region-id="921" left="36" top="2570.25" width="300" height="24" align-x="36" align-y="2580" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If you wished to postprocess this data without the outliers, simply <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">trim</c></inlineAttr></b></f> them from the dataset:</p></text></region><region region-id="1376" left="66" top="2601" width="21.75" height="18" align-x="78" align-y="2610" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>i</ml:id></ml:apply><ml:id>i</ml:id></ml:define></math><rendering item-idref="40" /></region><region region-id="1377" left="114" top="2601" width="86.25" height="12.75" align-x="137.25" align-y="2610" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>data</ml:id><ml:apply><ml:id>augment</ml:id><ml:id>x</ml:id><ml:id>y</ml:id></ml:apply></ml:define></math><rendering item-idref="41" /></region><region region-id="1378" left="222" top="2601" width="75.75" height="12.75" align-x="271.5" align-y="2610" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>rows</ml:id><ml:id>data</ml:id></ml:apply></ml:eval></math><rendering item-idref="42" /></region><region region-id="1368" left="66" top="2625" width="112.5" height="12.75" align-x="105.75" align-y="2634" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>datanew</ml:id><ml:apply><ml:id>trim</ml:id><ml:id>data</ml:id><ml:id>index</ml:id></ml:apply></ml:define></math><rendering item-idref="43" /></region><region region-id="1370" left="222" top="2625" width="92.25" height="12.75" align-x="288" align-y="2634" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>rows</ml:id><ml:id>datanew</ml:id></ml:apply></ml:eval></math><rendering item-idref="44" /></region><region region-id="1363" left="24" top="2646" width="335.25" height="195" align-x="24" align-y="2646" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="45" /><rendering item-idref="46" /></region><region region-id="1249" left="36" top="2845.5" width="117.75" height="12.75" align-x="36" align-y="2856" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Effects of outlier removal</p></text><indexes><phrase>effects of outlier removal</phrase><phrase>outlier removal, effects of</phrase></indexes></region><region region-id="1250" left="36" top="2870.25" width="346.5" height="36" align-x="36" align-y="2880" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The only reason to remove outliers from your calculation is if you expect them to skew your results. For example, the data above should be near-normally distributed.</p></text></region><region region-id="1251" left="66" top="2925" width="85.5" height="12.75" align-x="105" align-y="2934" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>mean</ml:id><ml:id>y</ml:id></ml:apply></ml:eval></math><rendering item-idref="47" /></region><region region-id="1349" left="192" top="2919.75" width="125.25" height="18" align-x="270.75" align-y="2934" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>mean</ml:id><ml:apply><ml:matcol /><ml:id>datanew</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:eval></math><rendering item-idref="48" /></region><region region-id="1252" left="66" top="2949" width="87.75" height="12.75" align-x="107.25" align-y="2958" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>Stdev</ml:id><ml:id>y</ml:id></ml:apply></ml:eval></math><rendering item-idref="49" /></region><region region-id="1350" left="192" top="2943.75" width="127.5" height="18" align-x="273" align-y="2958" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>Stdev</ml:id><ml:apply><ml:matcol /><ml:id>datanew</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:eval></math><rendering item-idref="50" /></region><region region-id="1257" left="36" top="2972.25" width="327" height="48" align-x="36" align-y="2982" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">While the mean, or center value, of the data, is practically the same without the outliers, the standard deviation grows smaller. 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