document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204909>Question 1204909</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Who's online? Tutor, please help.<BR>\n" );
document.write( "The mean and standard deviation of 15 observations are found to be 10 and 5 respectively. On rechecking, it was found that one of the observations with value 7 was incorrect. Calculate the correct mean and standard deviation if the correct observation is 22 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #841454 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=841454\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "mean = (sum of values)/(number of values)<BR>\n" );
document.write( "mean = S/n<BR>\n" );
document.write( "mean = S/15<BR>\n" );
document.write( "S/15 = 10<BR>\n" );
document.write( "S = 15*10<BR>\n" );
document.write( "S = 150\r<BR>\n" );
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document.write( "The value 7 is removed and replaced with 22.<BR>\n" );
document.write( "The net change for the sum is -7+22 = +15<BR>\n" );
document.write( "The sum 150 increases to S = 150+15 = 165\r<BR>\n" );
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document.write( "Correct mean = (sum of values)/(number of values)<BR>\n" );
document.write( "Correct mean = S/n<BR>\n" );
document.write( "Correct mean = 165/15<BR>\n" );
document.write( "<font color=red>Correct mean = 11</font>\r<BR>\n" );
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document.write( "Your teacher has not specified if you are using sample standard deviation or population standard deviation.<BR>\n" );
document.write( "I'll assume the 2nd option.\r<BR>\n" );
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document.write( "sigma = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma ALIGN=MIDDLE  ALT=\"sigma\"   DISABLED_style= \"max-width:90%; height:auto;\" > =  population standard deviation\r<BR>\n" );
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document.write( "One formula to calculate sigma is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma+=+sqrt%28+sum%28x%5E2%29%2Fn+-+%28sum%28x%29%2Fn%29%5E2+%29 ALIGN=MIDDLE  ALT=\"sigma+=+sqrt%28+sum%28x%5E2%29%2Fn+-+%28sum%28x%29%2Fn%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I'll replace the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28x%5E2%29 ALIGN=MIDDLE  ALT=\"sum%28x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > portion with A and replace the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28x%29 ALIGN=MIDDLE  ALT=\"sum%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > portion with S\r<BR>\n" );
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document.write( "The formula updates to <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma+=+sqrt%28+A%2Fn+-+%28S%2Fn%29%5E2+%29 ALIGN=MIDDLE  ALT=\"sigma+=+sqrt%28+A%2Fn+-+%28S%2Fn%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Notice the S/n part is really just the mean.\r<BR>\n" );
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document.write( "Let's determine the value of A based on the incorrect sigma = 5 value mentioned and the sample size n = 15<BR>\n" );
document.write( "The value of S/n is the old incorrect mean of 10 (do NOT use 11 just yet). <BR>\n" );
document.write( "This is because we want to peel back the layers to see what incorrect A value was found. From there we can make the proper adjustment.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma+=+sqrt%28+A%2Fn+-+%28S%2Fn%29%5E2+%29 ALIGN=MIDDLE  ALT=\"sigma+=+sqrt%28+A%2Fn+-+%28S%2Fn%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5+=+sqrt%28+A%2F15+-+%28150%2F15%29%5E2+%29 ALIGN=MIDDLE  ALT=\"5+=+sqrt%28+A%2F15+-+%28150%2F15%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5+=+sqrt%28+A%2F15+-+%2810%29%5E2+%29 ALIGN=MIDDLE  ALT=\"5+=+sqrt%28+A%2F15+-+%2810%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5E2+=+A%2F15+-+100 ALIGN=MIDDLE  ALT=\"5%5E2+=+A%2F15+-+100\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25+=+A%2F15+-+100 ALIGN=MIDDLE  ALT=\"25+=+A%2F15+-+100\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%2B100+=+A%2F15 ALIGN=MIDDLE  ALT=\"25%2B100+=+A%2F15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2F15+=+125 ALIGN=MIDDLE  ALT=\"A%2F15+=+125\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+15%2A125 ALIGN=MIDDLE  ALT=\"A+=+15%2A125\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+1875 ALIGN=MIDDLE  ALT=\"A+=+1875\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This indicates the incorrect sum of squares is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28x%5E2%29+=+1875 ALIGN=MIDDLE  ALT=\"sum%28x%5E2%29+=+1875\"   DISABLED_style= \"max-width:90%; height:auto;\" > based on the incorrect value of 7.\r<BR>\n" );
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document.write( "Remove the 7 and replace it with 22.<BR>\n" );
document.write( "We lose 7^2 from that sum of squares value, and gain 22^2.<BR>\n" );
document.write( "The net change is -7^2+22^2 = +435<BR>\n" );
document.write( "Meaning the 1875 should be 1875+435 = 2310 which is the correct sum of squares.\r<BR>\n" );
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document.write( "With the proper observation (22) in place, the sum of squares should be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28x%5E2%29+=+2310 ALIGN=MIDDLE  ALT=\"sum%28x%5E2%29+=+2310\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's now recalculate sigma based on that corrected value of A and corrected value of S (2310 and 165 respectively).<BR>\n" );
document.write( "In other words, the S/n part is 11 now. <BR>\n" );
document.write( "Refer to the previous section.<BR>\n" );
document.write( "The value n = 15 stays the same.\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma+=+sqrt%28+A%2Fn+-+%28S%2Fn%29%5E2+%29 ALIGN=MIDDLE  ALT=\"sigma+=+sqrt%28+A%2Fn+-+%28S%2Fn%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma+=+sqrt%28+2310%2F15+-+%28165%2F15%29%5E2+%29 ALIGN=MIDDLE  ALT=\"sigma+=+sqrt%28+2310%2F15+-+%28165%2F15%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma+=+sqrt%28+154+-+%2811%29%5E2+%29 ALIGN=MIDDLE  ALT=\"sigma+=+sqrt%28+154+-+%2811%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma+=+sqrt%28+154+-+121+%29 ALIGN=MIDDLE  ALT=\"sigma+=+sqrt%28+154+-+121+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma+=+sqrt%28+33+%29 ALIGN=MIDDLE  ALT=\"sigma+=+sqrt%28+33+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma+=+5.744562646 ALIGN=MIDDLE  ALT=\"sigma+=+5.744562646\"   DISABLED_style= \"max-width:90%; height:auto;\" > approximately<BR>\n" );
document.write( "Round this however your teacher instructs.\r<BR>\n" );
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document.write( "Answers:<BR>\n" );
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document.write( "mean = 11<BR>\n" );
document.write( "standard deviation = 5.744562646 approximately<BR>\n" );
document.write( "</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
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