document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196832>Question 1196832</A>: <I><SPAN STYLE=\"word-break: break-all;\"> PICTURE FOR PROBLEM  https://imagizer.imageshack.com/img924/482/OjKL0M.jpg\r<BR>\n" );
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document.write( "​a) Find the quartiles using technology.<BR>\n" );
document.write( "Q1= Q2=\r<BR>\n" );
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document.write( "​b) Find the quartiles using the Tukey method.<BR>\n" );
document.write( "Q1= Q2=\r<BR>\n" );
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document.write( "​c) Find the IQR using the quartiles from part b.\r<BR>\n" );
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document.write( "​d) Find the standard deviation.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #829849 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=829849\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "a)\r<BR>\n" );
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document.write( "The answer will slightly vary depending on which type of technology you use. Are you using a graphing calculator? Spreadsheet? A programming language such as R or Python?\r<BR>\n" );
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document.write( "Refer to this article for more details<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://en.wikipedia.org/wiki/Quartile\">https://en.wikipedia.org/wiki/Quartile</a>\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "Given Data<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >40.4</TD><TD >5.18</TD><TD >32.59</TD><TD >47.01</TD></tr><tr><TD >37.12</TD><TD >34.61</TD><TD >63.44</TD><TD >51.53</TD></tr><tr><TD >57.32</TD><TD >81.13</TD><TD >47.5</TD><TD >11.65</TD></tr><tr><TD >21.83</TD><TD >42.21</TD><TD >38.77</TD><TD >32.05</TD></tr><tr><TD >76.72</TD><TD >72.8</TD><TD >46.93</TD><TD >65.45</TD></tr>\n" );
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document.write( "Let's list out the sorted values from smallest to largest<BR>\n" );
document.write( "{5.18, 11.65, 21.83, 32.05, 32.59, 34.61, 37.12, 38.77, 40.4, 42.21, 46.93, 47.01, 47.5, 51.53, 57.32, 63.44, 65.45, 72.8, 76.72, 81.13}\r<BR>\n" );
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document.write( "There are n = 20 items in this data set<BR>\n" );
document.write( "The median is at the midpoint of slot 10 and 11<BR>\n" );
document.write( "since n/2 = 20/2 = 10\r<BR>\n" );
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document.write( "The items in the 10th and 11th slots are 42.21 and 46.93 respectively\r<BR>\n" );
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document.write( "The midpoint of which is (42.21+46.93)/2 = 44.57 and this is the median of the given set.<BR>\n" );
document.write( "The median is sometimes referred to as Q2\r<BR>\n" );
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document.write( "Let's form two new sets L and U<BR>\n" );
document.write( "L = lower set = items below the median 44.57<BR>\n" );
document.write( "U = upper set = items above the median 44.57\r<BR>\n" );
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document.write( "L = {5.18, 11.65, 21.83, 32.05, 32.59, 34.61, 37.12, 38.77, 40.4, 42.21}<BR>\n" );
document.write( "U = {46.93, 47.01, 47.5, 51.53, 57.32, 63.44, 65.45, 72.8, 76.72, 81.13}\r<BR>\n" );
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document.write( "Sets L and U have n/2 = 20/2 = 10 items each in them.\r<BR>\n" );
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document.write( "The median of set L is 33.60<BR>\n" );
document.write( "The median of set U is 60.38<BR>\n" );
document.write( "These are the values of Q1 and Q3 respectively.\r<BR>\n" );
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document.write( "Summary:<BR>\n" );
document.write( "Q1 = 33.60<BR>\n" );
document.write( "Q2 = 44.57 aka median<BR>\n" );
document.write( "Q3 = 60.38\r<BR>\n" );
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document.write( "c)\r<BR>\n" );
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document.write( "IQR = Q3 - Q1<BR>\n" );
document.write( "IQR = 60.38 - 33.60<BR>\n" );
document.write( "IQR = 26.78\r<BR>\n" );
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document.write( "d)\r<BR>\n" );
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document.write( "You can use a spreadsheet or graphing calculator to compute the standard deviation in one single command. <BR>\n" );
document.write( "So there isn't much for me to go over here if your teacher wants you to use technology.\r<BR>\n" );
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document.write( "However, I'll show a method to calculate the standard deviation by hand.\r<BR>\n" );
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document.write( "Check out this spreadsheet<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://docs.google.com/spreadsheets/d/1VsuY1rMMsdiF1GKgO_5x0Ts5vIOnn6hf5fhHlC4cIfw/edit?usp=sharing\">https://docs.google.com/spreadsheets/d/1VsuY1rMMsdiF1GKgO_5x0Ts5vIOnn6hf5fhHlC4cIfw/edit?usp=sharing</a><BR>\n" );
document.write( "The x column is the set of original values<BR>\n" );
document.write( "xbar = 45.312 is the sample mean (found by adding every x value up and dividing by n = 20) \r<BR>\n" );
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document.write( "The column labeled (x-xbar)^2 has us do these two things in the order mentioned<ol><li>Subtract xbar from each x value</li><li>Square the result of the previous step</li></ol>For example, <BR>\n" );
document.write( "(x-xbar)^2 = (5.18 - 45.312)^2 = 1610.577424<BR>\n" );
document.write( "and that result goes in the first entry of the (x-xbar)^2 column. It corresponds to x = 5.18\r<BR>\n" );
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document.write( "Once we calculate all the (x-xbar)^2 values, we add them all up to get <BR>\n" );
document.write( "Sum[ (x-xbar)^2 ] = 7843.32072<BR>\n" );
document.write( "In the spreadsheet provided above, I defined this value as the variable A.\r<BR>\n" );
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document.write( "Divide A over (n-1) to get the sample variance<BR>\n" );
document.write( "Sample Variance = A/(n-1) = 7843.32072/(20-1) = 412.8063537<BR>\n" );
document.write( "Then take the square root of that to get the sample standard deviation\r<BR>\n" );
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document.write( "SampleSD = sqrt(sampleVar)<BR>\n" );
document.write( "SampleSD = sqrt(412.8063537)<BR>\n" );
document.write( "SampleSD = 20.31763652\r<BR>\n" );
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document.write( "Or if you wanted the population variance, then,<BR>\n" );
document.write( "popVar = A/n = 7843.32072/20 = 392.166036<BR>\n" );
document.write( "and<BR>\n" );
document.write( "popSD = sqrt(popVar) = sqrt(392.166036) = 19.803182471512\r<BR>\n" );
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document.write( "So we have:<BR>\n" );
document.write( "sample standard deviation = 20.31763652<BR>\n" );
document.write( "population standard deviation = 19.803182471512<BR>\n" );
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