document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205795>Question 1205795</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the median of 80,80,80,80,80,85,85,85,85,85,90,90,90,90,90\r<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 #842835 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=842835\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer:  <font color=red size=4>85</font>\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "Method 1\r<BR>\n" );
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document.write( "The first task is to sort the values. Thankfully that has already been done for us. <BR>\n" );
document.write( "<u>Always</u> make sure the values are sorted before attempting to find the median.\r<BR>\n" );
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document.write( "The original string of values<BR>\n" );
document.write( "80,80,80,80,80,85,85,85,85,85,90,90,90,90,90<BR>\n" );
document.write( "rewrites to<BR>\n" );
document.write( "80,80,80,80,80,<BR>\n" );
document.write( "85,85,85,85,85,<BR>\n" );
document.write( "90,90,90,90,90<BR>\n" );
document.write( "which I think is easier to see how many items there are. This block of values is 3 items tall and 5 items across. There are 3*5 = 15 values total.\r<BR>\n" );
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document.write( "n = 15 is the sample size.<BR>\n" );
document.write( "n/2 = 15/2 = 7.5 which rounds to 8<BR>\n" );
document.write( "The median is in slot 8<BR>\n" );
document.write( "7 values are below it, and 7 values are above it<BR>\n" );
document.write( "7+1+7 = 15 values total\r<BR>\n" );
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document.write( "The value in slot 8 is 85. It's the 3rd copy of \"85\" in the second row of this block:<BR>\n" );
document.write( "80,80,80,80,80,<BR>\n" );
document.write( "85,85,<font color=red size=4>85</font>,85,85,<BR>\n" );
document.write( "90,90,90,90,90<BR>\n" );
document.write( "Or you can return back to the original string of values to count out 8 positions from the left.\r<BR>\n" );
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document.write( "Method 2\r<BR>\n" );
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document.write( "Chop off the first and last values of <BR>\n" );
document.write( "80,80,80,80,80,85,85,85,85,85,90,90,90,90,90<BR>\n" );
document.write( "to get<BR>\n" );
document.write( "<font color=red><s><font color=black>80</font></s></font>,80,80,80,80,85,85,85,85,85,90,90,90,90,<font color=red><s><font color=black>90</font></s></font><BR>\n" );
document.write( "which cleans up to<BR>\n" );
document.write( "80,80,80,80,85,85,85,85,85,90,90,90,90\r<BR>\n" );
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document.write( "Repeat the process again to go from<BR>\n" );
document.write( "80,80,80,80,85,85,85,85,85,90,90,90,90<BR>\n" );
document.write( "to<BR>\n" );
document.write( "80,80,80,85,85,85,85,85,90,90,90\r<BR>\n" );
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document.write( "Keep this process going until either (a) we end up with one value or (b) we end up with 2 values<BR>\n" );
document.write( "80,80,80,85,85,85,85,85,90,90,90<BR>\n" );
document.write( "80,80,85,85,85,85,85,90,90<BR>\n" );
document.write( "80,85,85,85,85,85,90<BR>\n" );
document.write( "85,85,85,85,85<BR>\n" );
document.write( "85,85,85<BR>\n" );
document.write( "85<BR>\n" );
document.write( "We see that <font color=red>85</font> is the middle most value.\r<BR>\n" );
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document.write( "Or what we can do is cross off say 3 items at a time from each endpoint. <BR>\n" );
document.write( "This is to help speed things up.<BR>\n" );
document.write( "80,80,80,80,80,85,85,85,85,85,90,90,90,90,90<BR>\n" );
document.write( "<font color=red><s><font color=black>80,80,80</font></s></font>,80,80,85,85,85,85,85,90,90,<font color=red><s><font color=black>90,90,90</font></s></font><BR>\n" );
document.write( "80,80,85,85,85,85,85,90,90<BR>\n" );
document.write( "<font color=red><s><font color=black>80,80,85</font></s></font>,85,85,85,<font color=red><s><font color=black>85,90,90</font></s></font><BR>\n" );
document.write( "85,<font color=red>85</font>,85\r<BR>\n" );
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document.write( "Note: If you end up with 2 values tied for the middle, then take the midpoint of them. This occurs only when the sample size is an even number. <BR>\n" );
document.write( "The midpoint formula is (p+q)/2 where p and q are the middle most values.\r<BR>\n" );
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document.write( "Method 3\r<BR>\n" );
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document.write( "Use software such as a speadsheet or online calculator.<BR>\n" );
document.write( "This method doesn't have much to explain about it, other than it's something I only recommend as a way to check your answer. <BR>\n" );
document.write( "It's best to know how to find the median by hand.<BR>\n" );
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