document.write( "Question 1173244: Consider a data set S with 51 signals of value 1 and 49 signals of value −1.
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document.write( "Calculate the mean x ̄ of S and the standard deviation σx.
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document.write( "What would happen to the mean and standard deviation if we added into our data
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document.write( "set a bunch of signals of value 0? \n" );
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Algebra.Com's Answer #798472 by Edwin McCravy(20056)![]() ![]() You can put this solution on YOUR website! Consider a data set S with 51 signals of value 1 and 49 signals of value −1. \n" ); document.write( "Calculate the mean x ̄ of S and the standard deviation σx. \n" ); document.write( " \r\n" ); document.write( "49 of the 1's would cancel all the -1's and leave two 1's, whose sum is 2.\r\n" ); document.write( "Divide that by 100 and the mean is 2/100 or 0.02\r\n" ); document.write( "\r\n" ); document.write( "Subtracting 0.02 from each of the 51 1's makes 51 0.98's. Subtracting 0.02 from\r\n" ); document.write( "each of the 49 -1's makes 49 -1.02's. Incidentally, if you added those\r\n" ); document.write( "together you'd just get 0. \r\n" ); document.write( "\r\n" ); document.write( "So we square the 51 0.98's and that gives us 51 0.9604's. We square the 49\r\n" ); document.write( "-1.02's and that gives us 49 1.0404's. Now we add 51*0.9604+49*1.0404 and get\r\n" ); document.write( "99.96. \r\n" ); document.write( "\r\n" ); document.write( "If the data set is that of a population we divide by 100 and get 0.9996 for the\r\n" ); document.write( "variance.\r\n" ); document.write( "\r\n" ); document.write( "If the data set is that of a sample we divide by 99 and get 1.00969697 for the\r\n" ); document.write( "variance. \r\n" ); document.write( "\r\n" ); document.write( "So if the data set is a population, we take the square root of 0.9996 and get\r\n" ); document.write( "0.99979998 for the standard deviation.\r\n" ); document.write( "\r\n" ); document.write( "If the data set is a sample, we take the square root of 1.00969697 and get\r\n" ); document.write( "1.004836788 for the standard deviation. \r\n" ); document.write( "\r\n" ); document.write( "What would happen to the mean and standard deviation if we added into our \n" ); document.write( "data set a bunch of signals of value 0? \r\n" ); document.write( "Suppose we add, say, 100 0's:\r\n" ); document.write( "\r\n" ); document.write( "49 of the 1's would still cancel all the -1's and leave two 1's, whose sum is\r\n" ); document.write( "2. The 0's would not change that. Divide that by 200 and the mean is 2/200 or\r\n" ); document.write( "0.01\r\n" ); document.write( "\r\n" ); document.write( "Subtracting 0.01 from each of the 51 1's makes 51 0.99's. Subtracting 0.01 from\r\n" ); document.write( "each of the 49 -1's makes 49 -1.01's. Subtracting 0.01 from each of the 100 0's\r\n" ); document.write( "makes 100 -0.01's. Incidentally, as before, and as in all such situations, if\r\n" ); document.write( "you added those together you'd just get 0. \r\n" ); document.write( "\r\n" ); document.write( "So we square the 51 0.99's and that gives us 51 0.9801's. We square the 49\r\n" ); document.write( "-1.01's and that gives us 49 1.0201's. We square the 100 -0.01's and that gives\r\n" ); document.write( "us 100 0.0001's. \r\n" ); document.write( "\r\n" ); document.write( "Now we add 51*0.9801+49*1.0201+100*(0.0001) and get 99.98. \r\n" ); document.write( "\r\n" ); document.write( "If the data set is that of a population we divide by 200 and get 0.9998 for the\r\n" ); document.write( "variance.\r\n" ); document.write( "\r\n" ); document.write( "If the data set is that of a sample we divide by 200 and get 0.4999 for the\r\n" ); document.write( "variance. \r\n" ); document.write( "\r\n" ); document.write( "So if the data set is a population, we take the square root of 0.9998 and get\r\n" ); document.write( "0.999899995 for the standard deviation.\r\n" ); document.write( "\r\n" ); document.write( "If the data set is a sample, we take the square root of 0.4999 and get\r\n" ); document.write( "0.707036067 for the standard deviation.\r\n" ); document.write( "\r\n" ); document.write( "------------\r\n" ); document.write( "\r\n" ); document.write( "My conclusion is that if we add as many 0's as we have data values, the mean\r\n" ); document.write( "would be about half as much and the standard deviation would be reduced by\r\n" ); document.write( "about 70%.\r\n" ); document.write( "\r\n" ); document.write( "You could try adding different amounts of zeros and seeing what you get.\r\n" ); document.write( "\r\n" ); document.write( "Edwin\n" ); document.write( " |