document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190761>Question 1190761</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The mean of a set of numbers is 500.<BR>\n" );
document.write( "a. If 10 is added to each of the numbers in the set, then what will be the mean of the new set?<BR>\n" );
document.write( "b. If each of the numbers in the set are multiplied by -5, then what will be the mean of the new <BR>\n" );
document.write( "set? </SPAN>\n" );
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document.write( "Let's say we had the set<BR>\n" );
document.write( "{400,500,600}<BR>\n" );
document.write( "The mean is 500. We can check as such<BR>\n" );
document.write( "(400+500+600)/3 = 1500/3 = 500\r<BR>\n" );
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document.write( "Now let's add 10 to each number<BR>\n" );
document.write( "{410,510,610}<BR>\n" );
document.write( "and repeat the process of finding the mean<BR>\n" );
document.write( "(410+510+610)/3 = 1530/3 = 510<BR>\n" );
document.write( "The mean has increased by 10.\r<BR>\n" );
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document.write( "We can rewrite that equation like so<BR>\n" );
document.write( "(410+510+610)/3 = (400+500+600 + 10+10+10)/3 <BR>\n" );
document.write( "(410+510+610)/3 = 500 + (10+10+10)/3 <BR>\n" );
document.write( "(410+510+610)/3 = 500 + (3*10)/3 <BR>\n" );
document.write( "(410+510+610)/3 = 500 + 10<BR>\n" );
document.write( "(410+510+610)/3 = 510<BR>\n" );
document.write( "If we add 10 to each number in the set, then the mean goes up by 10.\r<BR>\n" );
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document.write( "This can be applied in a more general sense, and it doesn't only work with this particular set (nor does it only apply to sets of 3 items)\r<BR>\n" );
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document.write( "{x1,x2,x3,...,xn} = original set of values<BR>\n" );
document.write( "S = sum of the original items = x1+x2+x3+...+xn<BR>\n" );
document.write( "S/n = original mean\r<BR>\n" );
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document.write( "Let's say we add on k to each number<BR>\n" );
document.write( "This is equivalent to adding nk because there are n copies of k being added on.<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "new mean = (S+nk)/n<BR>\n" );
document.write( "new mean = (S/n)+(nk/n)<BR>\n" );
document.write( "new mean = (old mean)+k<BR>\n" );
document.write( "Therefore, what happened with the \"adding 10\" example earlier wasn't just a coincidence.\r<BR>\n" );
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document.write( "Now let's consider multiplying each item in the set {x1,x2,...,xn} by some constant R<BR>\n" );
document.write( "We would get this new set {Rx1,Rx2,...,Rxn}\r<BR>\n" );
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document.write( "Let's find the new mean of this scaled set.<BR>\n" );
document.write( "new mean = (sum of the x values)/n <BR>\n" );
document.write( "new mean = (Rx1+Rx2+...+Rxn)/n <BR>\n" );
document.write( "new mean = (R*(x1+x2+...+xn))/n <BR>\n" );
document.write( "new mean = (R*S)/n<BR>\n" );
document.write( "new mean = R*(S/n) <BR>\n" );
document.write( "new mean = R*(old mean)\r<BR>\n" );
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document.write( "So if we were to multiply each value by R = -5, then,<BR>\n" );
document.write( "new mean = R*(old mean)<BR>\n" );
document.write( "new mean = -5*(500)<BR>\n" );
document.write( "new mean = -2500\r<BR>\n" );
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document.write( "Or you could go back to the example with 3 items from earlier<BR>\n" );
document.write( "{400,500,600}<BR>\n" );
document.write( "Multiply everything by -5 to get<BR>\n" );
document.write( "{-2000,-2500,-3000}<BR>\n" );
document.write( "Then compute the mean<BR>\n" );
document.write( "(-2000+(-2500)+(-3000))/3 = -7500/3 = -2500\r<BR>\n" );
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document.write( "To summarize:<BR>\n" );
document.write( "If you add 10 to each number, then the mean goes up by 10<BR>\n" );
document.write( "If you multiply each number by -5, then the mean is multiplied by -5\r<BR>\n" );
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document.write( "============================================<BR>\n" );
document.write( "Answers:<BR>\n" );
document.write( "a) 510<BR>\n" );
document.write( "b) -2500<BR>\n" );
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