document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=198809>Question 198809</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the following set of values for exercises 8–13.<BR>\n" );
document.write( "0.3  0.6  0.9  1.3  0.4  0.6  1.2  1.4  1.1  0.2  0.2\r<BR>\n" );
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document.write( "please Find the mean, median, and mode(s) of the data set. Round answers to the nearest hundredth, if necessary. \r<BR>\n" );
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document.write( "1.   Find the range.  \r<BR>\n" );
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document.write( "2.   Find the interquartile range.  \r<BR>\n" );
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document.write( "3.   Find the standard deviation. Round to the nearest hundredth.  \r<BR>\n" );
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document.write( "4.   Find the 25th and 75th percentiles.  \r<BR>\n" );
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document.write( "thanks for helping me out! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #149297 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=149297\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First sort the numbers into ascending order (from least to greatest):\r<BR>\n" );
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document.write( "0.2,0.2,0.3,0.4,0.6,0.6,0.9,1.1,1.2,1.3,1.4\r<BR>\n" );
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document.write( "Range:\r<BR>\n" );
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document.write( "To find the range, subtract the smallest number (in this case 0.2) from the largest number (in this case 1.4) like this:\r<BR>\n" );
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document.write( "So the range is 1.2\r<BR>\n" );
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document.write( "Interquartile Range:\r<BR>\n" );
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document.write( "First, find the median for the first half of the data set (ignore the median):\r<BR>\n" );
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document.write( "Median of 0.2, 0.2, 0.3, 0.4, 0.6 is 0.3 (since this is the middle most value)\r<BR>\n" );
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document.write( "Now find the median of the last half of the data set  (ignore the median):\r<BR>\n" );
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document.write( "Median of 0.9, 1.1, 1.2, 1.3, 1.4 is 1.2 (since this is the middle most value)\r<BR>\n" );
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document.write( "So the 1st and 3rd quartiles are 0.3 and 1.2 respectively.\r<BR>\n" );
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document.write( "This means that the interquartile range is \r<BR>\n" );
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document.write( "1.2 - 0.3 = 0.9\r<BR>\n" );
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document.write( "Note: the interquartile range is the difference between the 1st and 3rd quartiles\r<BR>\n" );
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document.write( "Note: I'm assuming that you want the sample standard deviation.\r<BR>\n" );
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document.write( "Standard Deviation:\r<BR>\n" );
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document.write( "Step 1) Find the mean:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE Mean=\frac{0.2+0.2+0.3+0.4+0.6+0.6+0.9+1.1+1.2+1.3+1.4}{11}=\frac{8.2}{11}=0.745454545454546\">\r<BR>\n" );
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document.write( "So the mean is roughly 0.75 (rounded to the nearest hundredth).\r<BR>\n" );
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document.write( "Step 2) Find the differences between the mean and every value:\r<BR>\n" );
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document.write( "Now subtract EVERY value from the mean (0.75) to get:\r<BR>\n" );
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document.write( "0.2-0.75 = -0.55, 0.2-0.75 = -0.55, 0.3-0.75 = -0.45, 0.4-0.75 = -0.35, 0.6-0.75 = -0.15, 0.6-0.75 = -0.15, 0.9-0.75 = 0.15, 1.1-0.75 = 0.35, 1.2-0.75 = 0.45, 1.3-0.75 = 0.55, 1.4-0.75 = 0.65, \r<BR>\n" );
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document.write( "So you should have a new column of values:   \r<BR>\n" );
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document.write( "-0.55, -0.55, -0.45, -0.35, -0.15, -0.15, 0.15, 0.35, 0.45, 0.55, 0.65, \r<BR>\n" );
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document.write( "Step 3) Squaring the differences:\r<BR>\n" );
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document.write( "Now square EVERY value in the last column to get: \r<BR>\n" );
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document.write( "(-0.55)^2 = 0.3, (-0.55)^2 = 0.3, (-0.45)^2 = 0.2, (-0.35)^2 = 0.12, (-0.15)^2 = 0.02, (-0.15)^2 = 0.02, (0.15)^2 = 0.02, (0.35)^2 = 0.12, (0.45)^2 = 0.2, (0.55)^2 = 0.3, (0.65)^2 = 0.42, \r<BR>\n" );
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document.write( "So you should have a new column of values:   \r<BR>\n" );
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document.write( "0.3, 0.3, 0.2, 0.12, 0.02, 0.02, 0.02, 0.12, 0.2, 0.3, 0.42, \r<BR>\n" );
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document.write( "Step 4) Adding up the squares:\r<BR>\n" );
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document.write( "Now add up the values of the last column:   \r<BR>\n" );
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document.write( "0.3 + 0.3 + 0.2 + 0.12 + 0.02 + 0.02 + 0.02 + 0.12 + 0.2 + 0.3 + 0.42 = 2.02\r<BR>\n" );
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document.write( "Step 5) Divide the sum by n-1:\r<BR>\n" );
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document.write( "Divide that sum by one less than the number of values in the list (in this case, it is n-1=11-1=10).\r<BR>\n" );
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document.write( "Step 6) Take the square root of the quotient:\r<BR>\n" );
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document.write( "Take the square root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.202 ALIGN=MIDDLE  ALT=\"0.202\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%280.202%29=0.449444101084885 ALIGN=MIDDLE  ALT=\"sqrt%280.202%29=0.449444101084885\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now round to the nearest hundredth to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.45 ALIGN=MIDDLE  ALT=\"0.45\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So the standard deviation is roughly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.45 ALIGN=MIDDLE  ALT=\"0.45\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "If you don't like that method, then here's another way:\r<BR>\n" );
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document.write( "First sort the numbers into ascending order (from least to greatest):\r<BR>\n" );
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document.write( "0.2,0.2,0.3,0.4,0.6,0.6,0.9,1.1,1.2,1.3,1.4\r<BR>\n" );
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document.write( "Standard Deviation: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE   \sigma=\sqrt{ \frac{1}{N-1}\displaystyle\sum_{i=0}^N (x_i-\bar{x})^2}\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \bar{x}\"> is the average, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x_i\"> is the ith number, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE N\"> is the number of numbers\r<BR>\n" );
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document.write( "So we can replace N with 11\r<BR>\n" );
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document.write( "Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11-1 ALIGN=MIDDLE  ALT=\"11-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 10\r<BR>\n" );
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document.write( "Replace  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \bar{x}\"> with 0.75\r<BR>\n" );
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document.write( "Expand the summation (replace each <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bi%5D ALIGN=MIDDLE  ALT=\"x%5Bi%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the respective number)\r<BR>\n" );
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document.write( "Subtract the terms in the parenthesis\r<BR>\n" );
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document.write( "Square each term\r<BR>\n" );
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document.write( "Add up all of the terms\r<BR>\n" );
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document.write( "Multiply\r<BR>\n" );
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document.write( "Take the square root\r<BR>\n" );
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document.write( "Round to the nearest hundredth\r<BR>\n" );
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document.write( "So the standard deviation is roughly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.45 ALIGN=MIDDLE  ALT=\"0.45\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "25th and 75th percentiles:\r<BR>\n" );
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document.write( "The 25th and 75th percentiles are simply the first and third quartiles. We found these earlier to be 0.3 and 1.2. \r<BR>\n" );
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document.write( "So the 25th and 75th percentiles are 0.3 and 1.2 </SPAN>\n" );
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