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align-x="6" align-y="6" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><png item-idref="2" /><link href="./dataTOC.xmcd" title="Links to Table of Contents" popup="false" /><rendering item-idref="3" /></region><region region-id="377" left="102" top="44.25" width="141.75" height="12" align-x="102" align-y="54" 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="Titre" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"><f family="Arial" charset="0"><u><inlineAttr line-through="false"><c val="#808000"><link href="./desc_stats.xmcd" title="Links to Section Contents" popup="false"><sp count="4" />DESCRIPTIVE STATISTICS</link></c></inlineAttr></u></f></p></text></region><region region-id="114" left="36" top="72.75" width="199.5" height="13.5" align-x="36" align-y="84" 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="Sous-titre" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Quantile-Quantile and Probability Plots</p></text><indexes><phrase>quantile-quantile and probability plots</phrase><phrase>descriptive statistics, quantile-quantile and probability plots</phrase><phrase>probability plots, and quantile-quantile plots</phrase></indexes></region><region region-id="107" left="66" top="97.5" width="153.75" height="12.75" align-x="66" align-y="108" 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="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">qqplot(v1,[v2 OR "distribution"])</p></text></region><region region-id="1282" left="36" top="122.25" width="312" height="72" align-x="36" align-y="132" 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="0" text-align="left" list-style-type="none" tabs="inherit">The quantile-quantile (q-q) plot is a plot of the <b><u><link href="./location.xmcd#quantiles" title="Links to section" popup="false">quantiles</link></u></b> of the first data set against the quantiles of the second data set, giving a visual measure of whether the two sets come from a common distribution. It is similar to a probability plot, where the quantiles for one of the data samples are replaced with the quantiles of a theoretical distribution. </p></text></region><region region-id="390" left="36" top="212.25" width="300.75" height="36" align-x="36" align-y="222" 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">Consider the following set of measured data of the strength of bars of bonded Si nitrate (column 3), manufactured over multiple labs and multiple batches.</p></text></region><region region-id="1331" left="30" top="267" width="334.5" height="132" align-x="55.5" align-y="276" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><component item-idref="4" disable-calc="false"><inputs hide="false" /><outputs hide="false"><ml:id>Data</ml:id></outputs></component><rendering item-idref="5" /></region><region region-id="714" left="36" top="410.25" width="338.25" height="48" align-x="36" align-y="420" 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">To investigate the relevance of the Batch on the results, we'll have to divide the data into two sets and compare them. To prepare the data, trim off the text labels, and pull out the two columns of interest, 2 and 3:</p></text></region><region region-id="1283" left="66" top="465" width="213" height="12.75" align-x="114" align-y="474" 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>BatchData</ml:id><ml:apply><ml:id>submatrix</ml:id><ml:id>Data</ml:id><ml:real>1</ml:real><ml:apply><ml:minus /><ml:apply><ml:id>rows</ml:id><ml:id>Data</ml:id></ml:apply><ml:real>1</ml:real></ml:apply><ml:real>2</ml:real><ml:real>3</ml:real></ml:apply></ml:define></math><rendering item-idref="6" /></region><region region-id="694" left="36" top="482.25" width="310.5" height="24" align-x="36" align-y="492" 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">Create two data vectors by Batch from the data in column 1 using <b><u><link href="./matching.xmcd" title="Links to section" popup="false">lookup functions</link></u></b>:</p></text></region><region region-id="743" left="66" top="519" width="141" height="12.75" align-x="99" align-y="528" 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>batch1</ml:id><ml:apply><ml:id>vlookup</ml:id><ml:real>1</ml:real><ml:id>BatchData</ml:id><ml:real>1</ml:real></ml:apply></ml:define></math><rendering item-idref="7" /></region><region region-id="744" left="66" top="543" width="141" height="12.75" align-x="99" align-y="552" 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>batch2</ml:id><ml:apply><ml:id>vlookup</ml:id><ml:real>2</ml:real><ml:id>BatchData</ml:id><ml:real>1</ml:real></ml:apply></ml:define></math><rendering item-idref="8" /></region><region region-id="726" left="0" top="570" width="6000" height="6" align-x="0" align-y="570" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="885" left="36" top="584.25" width="288" height="132" align-x="36" align-y="594" 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">Now plot a Q-Q plot, comparing the quantiles of one data set against those of the other. The output of the <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">qqplot</c></inlineAttr></b></f> function will be a three column matrix. The first column contains the quantiles of the first data set. The second column contains the quantiles of the second data set or standard probability distribution, and the third argument contains a least-squares-fit reference line to help visualize linearity of the data. It is assumed that the second and third columns will be plotted together on one axis (y), and the first column will be plotted on the other (x).</p></text></region><region region-id="894" left="66" top="729" width="144.75" height="12.75" align-x="111" align-y="738" 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>Batchplot</ml:id><ml:apply><ml:id>qqplot</ml:id><ml:id>batch2</ml:id><ml:id>batch1</ml:id></ml:apply></ml:define></math><rendering item-idref="9" /></region><region region-id="1309" left="222" top="729" width="123" height="12.75" align-x="231.75" align-y="738" 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>x</ml:id><ml:range><ml:apply><ml:id>min</ml:id><ml:id>batch2</ml:id></ml:apply><ml:apply><ml:id>max</ml:id><ml:id>batch2</ml:id></ml:apply></ml:range></ml:define></math><rendering item-idref="10" /></region><region region-id="898" left="36" top="752.25" width="243" height="12" align-x="36" align-y="762" 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">Calculate some quartiles for markers on the plot:</p></text></region><region region-id="895" left="66" top="777" width="132.75" height="12.75" align-x="97.5" align-y="786" 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>Q25_1</ml:id><ml:apply><ml:id>percentile</ml:id><ml:id>batch1</ml:id><ml:real>0.25</ml:real></ml:apply></ml:define></math><rendering item-idref="11" /></region><region region-id="900" left="222" top="777" width="132.75" height="12.75" align-x="253.5" align-y="786" 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>Q75_1</ml:id><ml:apply><ml:id>percentile</ml:id><ml:id>batch1</ml:id><ml:real>0.75</ml:real></ml:apply></ml:define></math><rendering item-idref="12" /></region><region region-id="903" left="66" top="795" width="132.75" height="12.75" align-x="97.5" align-y="804" 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>Q25_2</ml:id><ml:apply><ml:id>percentile</ml:id><ml:id>batch2</ml:id><ml:real>0.25</ml:real></ml:apply></ml:define></math><rendering item-idref="13" /></region><region region-id="904" left="222" top="795" width="132.75" height="12.75" align-x="253.5" align-y="804" 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>Q75_2</ml:id><ml:apply><ml:id>percentile</ml:id><ml:id>batch2</ml:id><ml:real>0.75</ml:real></ml:apply></ml:define></math><rendering item-idref="14" /></region><region region-id="1035" left="258" top="818.25" width="124.5" height="36" align-x="258" align-y="828" 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">Quantiles for batch 1 are shifted higher than they are for batch 2.</p></text></region><region region-id="1034" left="60" top="822.75" width="187.5" height="29.25" align-x="137.25" align-y="840" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:matrix rows="2" cols="2"><ml:id>Q25_1</ml:id><ml:id>Q25_2</ml:id><ml:id>Q75_1</ml:id><ml:id>Q75_2</ml:id></ml:matrix></ml:eval></math><rendering item-idref="15" /></region><region region-id="1328" left="36" top="858" width="346.5" height="282.75" align-x="36" align-y="858" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="16" /><rendering item-idref="17" /></region><region region-id="1329" left="36" top="1142.25" width="320.25" height="84" 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">Comparing the quantiles, or percentiles, of these two batches shows that they have not come from the same distribution, although the values for very small and very large quantiles are similar. A 1-to-1 reference line is plotted to emphasize how the data vary. We can also see that the Batch 1 quantiles are significantly higher than those for Batch 2, suggesting very different processing conditions.</p></text></region><region region-id="1327" left="0" top="1230" width="6000" height="6" align-x="0" align-y="1230" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="915" left="36" top="1255.5" width="74.25" height="12.75" align-x="36" align-y="1266" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag="probplot"><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">Probability plots</p></text><indexes><phrase>probability plots</phrase></indexes></region><region region-id="909" left="36" top="1274.25" width="321" height="72" align-x="43.5" align-y="1284" 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 <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">qqplot</c></inlineAttr></b></f> function can also be used to generate a normal probability plot, comparing a single data set against the quantiles for the normal distribution. This is yet another measure of <b><u><link href="./dispersion.xmcd" title="Links to section" popup="false">dispersion</link></u></b> in the data. The closer the points on this plot lie to the 45 degree reference line, the more normally distributed the given data is.</p></text></region><region region-id="1017" left="66" top="1353" width="111.75" height="46.5" align-x="91.5" align-y="1362" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><component item-idref="18" disable-calc="false"><inputs hide="false" /><outputs hide="false"><ml:id>Data</ml:id></outputs></component><rendering item-idref="19" /></region><region region-id="1019" left="36" top="1406.25" width="301.5" height="72" align-x="36" align-y="1416" 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">We are trying to determine the measurement process that resulted in this heat flow meter data were "in control," that is, if the data were drawn randomly and from a fixed distribution. We can check if the data are normally distributed by comparing them with the normal distribution in a probability plot:</p></text></region><region region-id="1145" left="66" top="1491" width="146.25" height="12.75" align-x="110.25" align-y="1500" 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>Normplot</ml:id><ml:apply><ml:id>qqplot</ml:id><ml:id>Data</ml:id><ml:str>normal</ml:str></ml:apply></ml:define></math><rendering item-idref="20" /></region><region region-id="1323" left="234" top="1490.25" width="122.25" height="13.5" align-x="234" align-y="1500" 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">equivalent to <region region-id="1324" left="303" top="1491" width="427.5" height="12.75" align-x="328.5" align-y="1500" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:apply><ml:id>qqplot</ml:id><ml:id>Data</ml:id></ml:apply></math><rendering item-idref="21" /></region></p></text></region><region region-id="1167" left="36" top="1514.25" width="295.5" height="36" align-x="36" align-y="1524" 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'll also calculate the slope and intercept of the <b><u><link href="./linreg.xmcd" title="Links to section" popup="false">line of best fit</link></u></b> to this probability plot, which is plotted to show how close to a straight line the quantiles are:</p></text></region><region region-id="1178" left="66" top="1566.75" width="159.75" height="29.25" align-x="93" align-y="1584" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:matrix rows="2" cols="1"><ml:id>b</ml:id><ml:id>m</ml:id></ml:matrix><ml:apply><ml:id>line</ml:id><ml:apply><ml:matcol /><ml:id>Normplot</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:matcol /><ml:id>Normplot</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="22" /></region><region region-id="1164" left="36" top="1608" width="321" height="223.5" align-x="36" align-y="1608" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="23" /><rendering item-idref="24" /></region><region region-id="1078" left="36" top="1838.25" width="295.5" height="48" align-x="36" align-y="1848" 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 very close correlation with the reference line would indicate that the samples are normally distributed. This check of normality will allow us to apply other criteria appropriately, such as tests for <b><u><link href="./outlier.xmcd" title="Links to section" popup="false">outliers</link></u></b>.</p></text></region><region region-id="1310" left="0" top="1896" width="6000" height="6" align-x="0" align-y="1896" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1140" left="36" top="1910.25" width="346.5" height="36" align-x="36" align-y="1920" 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">Here is a set of data that represents electrical insulation failure voltages for cables subjected to increasing voltage stress (<f family="Verdana" charset="0"><i><inlineAttr line-through="false"><c val="#000">Statistical Methods for Lifetime Data</c></inlineAttr></i></f>, by J.F. Lawless).</p></text></region><region region-id="1126" left="36" top="1953.75" width="477.75" height="18" align-x="48" align-y="1968" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:id>R</ml:id><ml:apply><ml:transpose /><ml:matrix rows="1" cols="20"><ml:real>39.4</ml:real><ml:real>45.3</ml:real><ml:real>49.2</ml:real><ml:real>49.4</ml:real><ml:real>51.3</ml:real><ml:real>52.0</ml:real><ml:real>53.2</ml:real><ml:real>53.2</ml:real><ml:real>54.9</ml:real><ml:real>55.5</ml:real><ml:real>57.1</ml:real><ml:real>57.2</ml:real><ml:real>57.5</ml:real><ml:real>59.2</ml:real><ml:real>61.0</ml:real><ml:real>62.4</ml:real><ml:real>63.8</ml:real><ml:real>64.3</ml:real><ml:real>67.3</ml:real><ml:real>67.7</ml:real></ml:matrix></ml:apply></ml:define></math><rendering item-idref="25" /></region><region region-id="1129" left="66" top="1983" width="32.25" height="12.75" align-x="74.25" align-y="1992" 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:real>8</ml:real></ml:range></ml:define></math><rendering item-idref="26" /></region><region region-id="1132" left="144" top="1983" width="63.75" height="18" align-x="169.5" align-y="1992" 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>bins</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:id>i</ml:id><ml:real>5</ml:real></ml:apply><ml:real>30</ml:real></ml:apply></ml:define></math><rendering item-idref="27" /></region><region region-id="1133" left="66" top="2007" width="82.5" height="12.75" align-x="90.75" align-y="2016" 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>stem</ml:id><ml:apply><ml:id>hist</ml:id><ml:id>bins</ml:id><ml:id>R</ml:id></ml:apply></ml:define></math><rendering item-idref="28" /></region><region region-id="1135" left="60" top="2034" width="237.75" height="211.5" align-x="60" align-y="2034" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="29" /><rendering item-idref="30" /></region><region region-id="1142" left="36" top="2228.25" width="342" height="36" align-x="36" align-y="2238" 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 histogram reveals that the data is not normal in distribution. It's skewed to one side. Let's compare its quantiles with those for the Weibull distribution:</p></text></region><region region-id="1268" left="66" top="2277" width="134.25" height="12.75" align-x="109.5" align-y="2286" 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>Weibplot</ml:id><ml:apply><ml:id>qqplot</ml:id><ml:id>R</ml:id><ml:str>weibull</ml:str></ml:apply></ml:define></math><rendering item-idref="31" /></region><region region-id="1269" left="66" top="2304.75" width="158.25" height="29.25" align-x="93" align-y="2322" 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:matrix rows="2" cols="1"><ml:id>b</ml:id><ml:id>m</ml:id></ml:matrix><ml:apply><ml:id>line</ml:id><ml:apply><ml:matcol /><ml:id>Weibplot</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:matcol /><ml:id>Weibplot</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="32" /></region><region region-id="1267" left="36" top="2348.25" width="330" height="24" align-x="36" align-y="2358" 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 the Weibull parameters of interest are approximately given by the slope of the line</p></text></region><region region-id="1258" left="66" top="2384.25" width="26.25" height="13.5" align-x="78" align-y="2394" 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>β</ml:id><ml:id>m</ml:id></ml:define></math><rendering item-idref="33" /></region><region region-id="1270" left="144" top="2384.25" width="62.25" height="13.5" align-x="155.25" align-y="2394" 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>β</ml:id></ml:eval></math><rendering item-idref="34" /></region><region region-id="1280" left="36" top="2408.25" width="97.5" height="12" align-x="36" align-y="2418" 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">and the x-intercept</p></text></region><region region-id="1273" left="66" top="2427.75" width="186" height="18" align-x="78.75" align-y="2442" 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:apply><ml:id>exp</ml:id><ml:apply><ml:id>intercept</ml:id><ml:apply><ml:matcol /><ml:id>Weibplot</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:matcol /><ml:id>Weibplot</ml:id><ml:real>0</ml:real></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="35" /></region><region region-id="1281" left="66" top="2456.25" width="63" height="13.5" align-x="78" align-y="2466" 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="36" /></region><region region-id="1313" left="0" top="2478" width="6000" height="6" align-x="0" align-y="2478" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1312" left="36" top="2492.25" width="314.25" height="24" align-x="36" align-y="2502" 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">Here is the plot. When the Weibull quantiles are used, the x axis is returned in units of ln(quantile) for the original data set.</p></text></region><region region-id="1262" left="60" top="2532" width="276.75" height="252.75" align-x="60" align-y="2532" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="37" /><rendering item-idref="38" /></region><region region-id="1279" left="6" top="2790" width="427.5" height="5.25" align-x="6" align-y="2790" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><png item-idref="39" /><rendering item-idref="40" /></region></regions><binaryContent><item item-id="1" content-encoding="gzip">H4sIAAAAAAAA/4zMuw6CMBgF4P5tuYoyOf8mLjZh4hmIk4mPQErFW0I0yODb66HgbpvTy9fm
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