document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1158372>Question 1158372</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the range, standard deviation, and variance of <BR>\n" );
document.write( "A survey of 10 fast-food restaurant noted the number of calories in a mid-sized hamburger. The following are results <BR>\n" );
document.write( "514 507 502 498 496 506 458 478 463 514  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854591 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5396)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854591\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Range, standard deviation, and variance are measures of the dispersion of the data in a set.<BR>\n" );
document.write( "A long time ago, in college, I would calculate them using paper and pen, based on their definitions.<BR>\n" );
document.write( "Nowadays, a calculator, or computer software can calculate them for us easily, but we have to understand what this measure tell us and exactly what our devices.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Range:<BR>\n" );
document.write( "The range is the difference between the smallest and greatest values.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Range=514-458=highlight%2856%29 ALIGN=MIDDLE  ALT=\"Range=514-458=highlight%2856%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Variance:<BR>\n" );
document.write( "The mean (average) of all 10 values is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28514%2B507%2B502%2B498%2B496%2B506%2B458%2B478%2B463%2B514%29%2F10=4936%2F10=493.6 ALIGN=MIDDLE  ALT=\"%28514%2B507%2B502%2B498%2B496%2B506%2B458%2B478%2B463%2B514%29%2F10=4936%2F10=493.6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The deviations from the mean for the individual values are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=514-493.6=20.4 ALIGN=MIDDLE  ALT=\"514-493.6=20.4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (514 appears twice)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=507-493.6=13.4 ALIGN=MIDDLE  ALT=\"507-493.6=13.4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=502-493.6=8.4 ALIGN=MIDDLE  ALT=\"502-493.6=8.4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=498-493.6=4.4 ALIGN=MIDDLE  ALT=\"498-493.6=4.4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=496-493.6=2.4 ALIGN=MIDDLE  ALT=\"496-493.6=2.4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=506-493.6=12.4 ALIGN=MIDDLE  ALT=\"506-493.6=12.4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=458-493.6=-35.6%7D%7D%0D%0A%7B%7B%7B478-493.6=-15.6 ALIGN=MIDDLE  ALT=\"458-493.6=-35.6%7D%7D%0D%0A%7B%7B%7B478-493.6=-15.6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=463-493.6=-30.6 ALIGN=MIDDLE  ALT=\"463-493.6=-30.6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Their squares are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20.4%5E2=416.16 ALIGN=MIDDLE  ALT=\"20.4%5E2=416.16\"   DISABLED_style= \"max-width:90%; height:auto;\" > (514 appears twice)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13.4%5E2=179.56 ALIGN=MIDDLE  ALT=\"13.4%5E2=179.56\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8.4%5E2=70.56 ALIGN=MIDDLE  ALT=\"8.4%5E2=70.56\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4.4%5E2=19.36 ALIGN=MIDDLE  ALT=\"4.4%5E2=19.36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.4%5E2=5.76 ALIGN=MIDDLE  ALT=\"2.4%5E2=5.76\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12.4%5E2=153.76 ALIGN=MIDDLE  ALT=\"12.4%5E2=153.76\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-35.6%5E2=1267.36%7D%7D%0D%0A%7B%7B%7B-15.6%5E2=243.36 ALIGN=MIDDLE  ALT=\"-35.6%5E2=1267.36%7D%7D%0D%0A%7B%7B%7B-15.6%5E2=243.36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-30.6%5E2=936.36 ALIGN=MIDDLE  ALT=\"-30.6%5E2=936.36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The variance is the average of those squares: <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28416.16%2B416.16%2B179.56%2B70.56%2B19.36%2B5.76%2B153.76%2B1267.36%2B243.36%2B936.36%29%2F10=3708.4%2F10=highlight%28370.84%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Standard deviation:<BR>\n" );
document.write( "The standard deviation, as asked in this question, is the square root of the variance as calculated above:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28370.84%29=highlight%2819.26%29 ALIGN=MIDDLE  ALT=\"sqrt%28370.84%29=highlight%2819.26%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded)<BR>\n" );
document.write( "Software that calculates statistics would have to forms of standard deviation:<BR>\n" );
document.write( "the standard deviation of a population, calculated as above, and<BR>\n" );
document.write( "the standard deviation of a sample, used to estimate the dispersion of the whole population of data based on a sample of N items. In the second case, the sum of squares in divided by N-1 instead. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );