document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1037849>Question 1037849</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If the standard deviation of 3 numbers (a), (2a+1&#65289;and (2) is sqr(1/2), find the value of a.\r<BR>\n" );
document.write( "\n" );
document.write( "Any help appreciated! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #652490 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=652490\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here is a general outline of how to find the sample standard deviation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Step 1) Add up all the values<BR>\n" );
document.write( "Step 2) Divide the sum in step 1 by three (there are three values added up). This is the mean M<BR>\n" );
document.write( "Step 3) Subtract each value from the mean M<BR>\n" );
document.write( "Step 4) Square each result in step 3<BR>\n" );
document.write( "Step 5) Add up the reslts in step 4<BR>\n" );
document.write( "Step 6) Divide the result in step 5 by n-1 = 3-1 = 2<BR>\n" );
document.write( "Step 7) Take the square root of the result of step 6\r<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Let's go through all the steps shown in the outline above\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Step 1) \r<BR>\n" );
document.write( "\n" );
document.write( "(value1)+(value2)+(value3) = (a)+(2a+1)+(2) <BR>\n" );
document.write( "(value1)+(value2)+(value3) = 3a+3\r<BR>\n" );
document.write( "\n" );
document.write( "The result of step 1 is 3a+3\r<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Step 2)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We divide the result from step 1 (which was 3a+3) by 3 because there are 3 values\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(result in step 1)/3 = (3a+3)/3 = a+1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the mean M is a+1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "M = a+1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We will use this in step 3\r<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Step 3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Subtract each value from the mean M\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(value1) - (mean) = (a) - (a+1) = a-a-1 = -1<BR>\n" );
document.write( "(value2) - (mean) = (2a+1) - (a+1) = 2a+1-a-1 = a<BR>\n" );
document.write( "(value3) - (mean) = (2) - (a+1) = 2-a-1 = 1-a\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The results after the subtractions are:<BR>\n" );
document.write( "-1<BR>\n" );
document.write( "a<BR>\n" );
document.write( "1-a\r<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Step 4)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The results of step 3 were -1, a, and 1-a. Let's square the results\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(previous result1)^2 = (-1)^2 = 1<BR>\n" );
document.write( "(previous result2)^2 = (a)^2 = a^2<BR>\n" );
document.write( "(previous result3)^2 = (1-a)^2 = 1-2a+a^2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The results of step 4 are:<BR>\n" );
document.write( "1<BR>\n" );
document.write( "a^2<BR>\n" );
document.write( "1-2a+a^2\r<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Step 5)\r<BR>\n" );
document.write( "\n" );
document.write( "Now let's add up the results of the previous step (step 4)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(previous result1)+(previous result2)+(previous result3) = (1)+(a^2)+(1-2a+a^2)\r<BR>\n" );
document.write( "\n" );
document.write( "(previous result1)+(previous result2)+(previous result3) = 2a^2-2a+2\r<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Step 6) Divide the previous result by 2 (n-1 = 3-1 = 2)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(previous result)/2 = (2a^2-2a+2)/2 = a^2-a+1\r<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Step 7) \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Take the square root of the previous result and we get\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%5E2-a%2B1%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%5E2-a%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "========================================================================================================<BR>\n" );
document.write( "========================================================================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "That is a lot of work, but we found the sample standard deviation of the three values to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%5E2-a%2B1%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%5E2-a%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Set this equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281%2F2%29 ALIGN=MIDDLE  ALT=\"sqrt%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and solve for 'a'\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%5E2-a%2B1%29=sqrt%281%2F2%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%5E2-a%2B1%29=sqrt%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28a%5E2-a%2B1%29%29%5E2=%28sqrt%281%2F2%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%28a%5E2-a%2B1%29%29%5E2=%28sqrt%281%2F2%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-a%2B1=1%2F2 ALIGN=MIDDLE  ALT=\"a%5E2-a%2B1=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2-2a%2B2=1 ALIGN=MIDDLE  ALT=\"2a%5E2-2a%2B2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2-2a%2B2-1=0 ALIGN=MIDDLE  ALT=\"2a%5E2-2a%2B2-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2-2a%2B1=0 ALIGN=MIDDLE  ALT=\"2a%5E2-2a%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then I run into a problem: if I use the quadratic formula to solve for 'a', I get two non-real answers. So it's making me think that there is a typo somewhere in the problem. You'll have to ask your teacher about it.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Edit: I redid the problem but instead of dividing by n-1 = 2 (in step 6), I divided by n = 3. This is to reflect the population standard deviation and not the sample standard deviation. Every other step is the same. Doing that leads to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%282%2B2a%5E2-2a%29%2F3%29+=+sqrt%281%2F2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%282%2B2a%5E2-2a%29%2F3%29+=+sqrt%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which leads to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+1%2F2 ALIGN=MIDDLE  ALT=\"a+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. However, I'm not sure which form of standard deviation your teacher wants.  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );