document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=196717>Question 196717</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how do you find the variance and standard of the following data? 12,8,13,9,13,11,12,11,9 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #147570 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=147570\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Standard Deviation:\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( "8,9,9,11,11,12,12,13,13\r<BR>\n" );
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document.write( "Now let's find the mean:\r<BR>\n" );
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document.write( "To find the mean, add up all of the numbers and divide the sum by the number of numbers (which in this case is 9).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Mean=%288%2B9%2B9%2B11%2B11%2B12%2B12%2B13%2B13%29%2F9=98%2F9=10.8888888888889 ALIGN=MIDDLE  ALT=\"Mean=%288%2B9%2B9%2B11%2B11%2B12%2B12%2B13%2B13%29%2F9=98%2F9=10.8888888888889\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the mean is 10.8888888888889\r<BR>\n" );
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document.write( "Now let's find the standard deviation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE  \sigma=\sqrt{ \frac{1}{N}\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 arithmetic mean, <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 9 (since there are 9 numbers)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE\sqrt{ \frac{1}{9}\displaystyle\sum_{i=0}^{9} (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 10.8888888888889\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE\sqrt{ \frac{1}{9}\displaystyle\sum_{i=0}^{9} (x_i-10.8888888888889)^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%2F9%29%28%288-10.8888888888889%29%5E2%2B%289-10.8888888888889%29%5E2%2B%289-10.8888888888889%29%5E2%2B%2811-10.8888888888889%29%5E2%2B%2811-10.8888888888889%29%5E2%2B%2812-10.8888888888889%29%5E2%2B%2812-10.8888888888889%29%5E2%2B%2813-10.8888888888889%29%5E2%2B%2813-10.8888888888889%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%2F9%29%28%28-2.889%29%5E2%2B%28-1.889%29%5E2%2B%28-1.889%29%5E2%2B%280.111%29%5E2%2B%280.111%29%5E2%2B%281.111%29%5E2%2B%281.111%29%5E2%2B%282.111%29%5E2%2B%282.111%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( "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%2F9%2926.888889%29 ALIGN=MIDDLE  ALT=\"sqrt%28%281%2F9%2926.888889%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%282.98765433333333%29 ALIGN=MIDDLE  ALT=\"sqrt%282.98765433333333%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=1.7284832464717 ALIGN=MIDDLE  ALT=\"1.7284832464717\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the standard deviation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.7284832464717 ALIGN=MIDDLE  ALT=\"1.7284832464717\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sigma = 1.7284832464717\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sigma ^2\">  is the variance, this means that the variance is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sigma ^2 = 1.7284832464717^2 = 2.98765433333334\"> </SPAN>\n" );
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