document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=89828>Question 89828</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me find the standard deviation of the data values:\r<BR>\n" );
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document.write( "10, 12, 12, 15, 16, 16, 16, 17, 19, 20  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #65209 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=65209\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First 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 10).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Mean=%2810%2B+12%2B+12%2B+15%2B+16%2B+16%2B+16%2B+17%2B+19%2B+20%29%2F10=153%2F10=15.3 ALIGN=MIDDLE  ALT=\"Mean=%2810%2B+12%2B+12%2B+15%2B+16%2B+16%2B+16%2B+17%2B+19%2B+20%29%2F10=153%2F10=15.3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the mean is 15.3\r<BR>\n" );
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document.write( "Now use this formula to find the standard deviation:\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}\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 10\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE\sqrt{ \frac{1}{10}\displaystyle\sum_{i=0}^{10} (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 15.3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE\sqrt{ \frac{1}{10}\displaystyle\sum_{i=0}^{10} (x_i-15.3)^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%2F10%29%28%2810-15.3%29%5E2%2B%28+12-15.3%29%5E2%2B%28+12-15.3%29%5E2%2B%28+15-15.3%29%5E2%2B%28+16-15.3%29%5E2%2B%28+16-15.3%29%5E2%2B%28+16-15.3%29%5E2%2B%28+17-15.3%29%5E2%2B%28+19-15.3%29%5E2%2B%28+20-15.3%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%2F10%29%28%28-5.3%29%5E2%2B%28-3.3%29%5E2%2B%28-3.3%29%5E2%2B%28-0.3%29%5E2%2B%280.7%29%5E2%2B%280.7%29%5E2%2B%280.7%29%5E2%2B%281.7%29%5E2%2B%283.7%29%5E2%2B%284.7%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%2F10%29%2828.09%2B10.89%2B10.89%2B0.09%2B0.49%2B0.49%2B0.49%2B2.89%2B13.69%2B22.09%29%29 ALIGN=MIDDLE   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%2F10%2990.1%29 ALIGN=MIDDLE  ALT=\"sqrt%28%281%2F10%2990.1%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%289.01%29 ALIGN=MIDDLE  ALT=\"sqrt%289.01%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=3.00166620396073 ALIGN=MIDDLE  ALT=\"3.00166620396073\"   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=3.00166620396073 ALIGN=MIDDLE  ALT=\"3.00166620396073\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "You can verify this with any calculator that has stats functions. </SPAN>\n" );
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