document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201889>Question 1201889</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two data sets have the same range, can you assume the interquartile ranges are about the same? Please include an example to justify your answer.<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 #836449 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</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=836449\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Consider this seven element list<BR>\n" );
document.write( "{1,2,3,4,5,6,7}\r<BR>\n" );
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document.write( "The median is 4 because it's the middle most value.<BR>\n" );
document.write( "This can be seen fairly quickly, or we could cross off the first and last terms (1 and 7) to get {2,3,4,5,6}<BR>\n" );
document.write( "Then repeat again to get {3,4,5} and at this point it's more clear 4 is at the very middle.\r<BR>\n" );
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document.write( "What's another way to find the median?<BR>\n" );
document.write( "We have n = 7 items. <BR>\n" );
document.write( "The midpoint is at slot 4 because (n+1)/2 = (7+1)/2 = 8/2 = 4<BR>\n" );
document.write( "Or you could say n/2 = 7/2 = 3.5 which rounds to 4.<BR>\n" );
document.write( "Either formula only works when n is odd.\r<BR>\n" );
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document.write( "We'll use the median to split the data into two halves<BR>\n" );
document.write( "Split the data into a lower set L and upper set U<BR>\n" );
document.write( "L = lower set<BR>\n" );
document.write( "L = stuff smaller than the median<BR>\n" );
document.write( "L = {1,2,3}<BR>\n" );
document.write( "U = upper set<BR>\n" );
document.write( "U = stuff larger than the median<BR>\n" );
document.write( "U = {5,6,7}<BR>\n" );
document.write( "The median is not part of either subset.\r<BR>\n" );
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document.write( "The median of set L is 2, which is the first quartile Q1.<BR>\n" );
document.write( "The median of set U is 6, which is the third quartile Q3.\r<BR>\n" );
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document.write( "Q1 = 2<BR>\n" );
document.write( "Q3 = 6<BR>\n" );
document.write( "IQR = interquartile range<BR>\n" );
document.write( "IQR = Q3 - Q1<BR>\n" );
document.write( "IQR = 6-2<BR>\n" );
document.write( "IQR = 4\r<BR>\n" );
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document.write( "And,<BR>\n" );
document.write( "range = max - min<BR>\n" );
document.write( "range = 7 - 1<BR>\n" );
document.write( "range = 6\r<BR>\n" );
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document.write( "Summary: The data set {1,2,3,4,5,6,7} has range = 6 and IQR = 4.\r<BR>\n" );
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document.write( "-----------------------\r<BR>\n" );
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document.write( "We can add some constant to each item of that set to shift things over.<BR>\n" );
document.write( "I'll add 10 to each value<BR>\n" );
document.write( "{1,2,3,4,5,6,7} turns into {11,12,13,14,15,16,17}\r<BR>\n" );
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document.write( "The range is still 6 because max-min = 17-11 = 6<BR>\n" );
document.write( "The IQR is still 4 because Q3 - Q1 = 16-12 = 4<BR>\n" );
document.write( "This demonstrates that shifting the values the same amount won't affect the range, and won't change the IQR either.\r<BR>\n" );
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document.write( "But what if we replaced 12 and 16 with say 12.5 and 15.5 respectively?<BR>\n" );
document.write( "What if we had {11,<font color=red>12.5</font>,13,14,15,<font color=red>15.5</font>,17}<BR>\n" );
document.write( "Again the range is still 6 because max-min = 17-11 = 6\r<BR>\n" );
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document.write( "But the IQR is now different<BR>\n" );
document.write( "IQR = Q3 - Q1 = 15.5-12.5 = 3<BR>\n" );
document.write( "The IQR is smaller since I moved Q1 and Q3 closer together.<BR>\n" );
document.write( "Earlier we had IQR = 4.\r<BR>\n" );
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document.write( "Here is a box plot, aka box-and-whisker plot, of {11,12,13,14,15,16,17} in blue<BR>\n" );
document.write( "Compared to a box plot of {11,12.5,13,14,15,15.5,17} in red.<BR>\n" );
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document.write( "The distance from left whisker (11) to right whisker (17) is the same for both box plots. <BR>\n" );
document.write( "That distance is the range 6.<BR>\n" );
document.write( "However, the width of the boxes themselves aren't the same. <BR>\n" );
document.write( "The red box at the bottom is slightly more skinny compared to the blue box up top.<BR>\n" );
document.write( "Q1 = left edge of the box<BR>\n" );
document.write( "Q3 = right edge of the box<BR>\n" );
document.write( "IQR = Q3 - Q1 = distance from Q1 to Q3 = width of the box (ignore whiskers)\r<BR>\n" );
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document.write( "Conclusion: Two data sets having the same range does NOT guarantee the IQR's will be the same.<BR>\n" );
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