document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1018958>Question 1018958</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the standard deviation first 13 natural numbers? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #634933 by </B><A HREF=/tutors/aboutme.mpl?userid=Marz157><B>Marz157(7)</B></A><img src=\"/images/stars/stars-05.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Marz157><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=634933\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'm going to assume that your definition of Natural numbers starts a 1, so the first 13 are the numbers from 1 to 13. \r<BR>\n" );
document.write( "\n" );
document.write( "Also I'm assuming that you are getting the population standard deviation which means we will use N instead of N-1 later on.\r<BR>\n" );
document.write( "\n" );
document.write( "So for standard deviation we will first need the mean (also called the average). For that we need to sum all the numbers and divide by how many numbers we added up (in this case 13).\r<BR>\n" );
document.write( "\n" );
document.write( "There is a nice trick for summing up numbers starting at 1 which is\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+%2A+%28n%2B1%29+%2F+2 ALIGN=MIDDLE  ALT=\"n+%2A+%28n%2B1%29+%2F+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which gives us 91 for n=13. When we divide by n to get the average, we get 7. \r<BR>\n" );
document.write( "\n" );
document.write( "Standard deviation is defined as the square root of the sum of the squared errors divided by n (or sometimes n-1, check with your teacher for what they want). \r<BR>\n" );
document.write( "\n" );
document.write( "The \"error\" means the difference from the mean. So for example the error for the first number 1 is (7-1 = 6). We square this to get 36.\r<BR>\n" );
document.write( "\n" );
document.write( "We do this for each number and add them up.\r<BR>\n" );
document.write( "\n" );
document.write( "1) 7-1 = 6, 6*2 = 36<BR>\n" );
document.write( "2) 7-2 = 5, 5^2 = 25<BR>\n" );
document.write( "3) ... 16<BR>\n" );
document.write( "4) ... 9<BR>\n" );
document.write( "5) ... 4<BR>\n" );
document.write( "6) ... 1<BR>\n" );
document.write( "7) ... 0<BR>\n" );
document.write( "8) ... 1<BR>\n" );
document.write( "9) ... 4<BR>\n" );
document.write( "10) ... 9<BR>\n" );
document.write( "11) ... 16<BR>\n" );
document.write( "12) ... 25<BR>\n" );
document.write( "13) ... 36\r<BR>\n" );
document.write( "\n" );
document.write( "Summing those we get 182. We divide 182 by 13 since we have 13 numbers = 14. Lastly we take the square root to get 3.7416\r<BR>\n" );
document.write( "\n" );
document.write( "Wikipedia has a great example of computing a standard deviation https://en.wikipedia.org/wiki/Standard_deviation#Basic_examples </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );