document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=715886>Question 715886</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Variance and standard deviation for 7, 8, 9, 6, 9, 12, 22, 15 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #439643 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=439643\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Mean=%286%2B7%2B8%2B9%2B9%2B12%2B15%2B22%29%2F8=88%2F8=11 ALIGN=MIDDLE  ALT=\"Mean=%286%2B7%2B8%2B9%2B9%2B12%2B15%2B22%29%2F8=88%2F8=11\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xbar+=+11 ALIGN=MIDDLE  ALT=\"xbar+=+11\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Sample 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 8\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE\sqrt{ \frac{1}{8-1}\displaystyle\sum_{i=0}^8 (x_i-\bar{x})^2}\">\r<BR>\n" );
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document.write( "Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8-1 ALIGN=MIDDLE  ALT=\"8-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 7\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE\sqrt{ \frac{1}{7}\displaystyle\sum_{i=0}^8 (x_i-\bar{x})^2}\">\r<BR>\n" );
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document.write( "Replace  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \bar{x}\"> with 11\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE\sqrt{ \frac{1}{7}\displaystyle\sum_{i=0}^8 (x_i-11)^2}\">\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( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%281%2F7%29%28%286-11%29%5E2%2B%287-11%29%5E2%2B%288-11%29%5E2%2B%289-11%29%5E2%2B%289-11%29%5E2%2B%2812-11%29%5E2%2B%2815-11%29%5E2%2B%2822-11%29%5E2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Subtract the terms in the parenthesis\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%281%2F7%29%28%28-5%29%5E2%2B%28-4%29%5E2%2B%28-3%29%5E2%2B%28-2%29%5E2%2B%28-2%29%5E2%2B%281%29%5E2%2B%284%29%5E2%2B%2811%29%5E2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Square each term\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%281%2F7%29%2825%2B16%2B9%2B4%2B4%2B1%2B16%2B121%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%281%2F7%29%2825%2B16%2B9%2B4%2B4%2B1%2B16%2B121%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Add up all of the terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%281%2F7%29196%29 ALIGN=MIDDLE  ALT=\"sqrt%28%281%2F7%29196%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2828%29 ALIGN=MIDDLE  ALT=\"sqrt%2828%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Take the square root\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5.29150262212918 ALIGN=MIDDLE  ALT=\"5.29150262212918\"   DISABLED_style= \"max-width:90%; height:auto;\" > This is is the sample standard deviation\r<BR>\n" );
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document.write( "The sample variance is simply the square of the sample standard deviation, so the sample variance is 28. </SPAN>\n" );
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