document.write( "Question 457977: A certain set of data has a mean of 20 and a Standard deviation of 4.5. It is found that two pieces of data have been omitted, one of value 30 and the other of value 25. Find the mean and S.D of the full 25 pieces of data. \n" ); document.write( "

Algebra.Com's Answer #314185 by Edwin McCravy(20054) $\"\"$ $\"About$

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document.write( "The other tutor's answer for the correct standard deviation is wrong.\r\n" );
document.write( "Also he used the formula for the standard deviation of a sample, and\r\n" );
document.write( "this student has informed me that this is to be considered as an \r\n" );
document.write( "entire population, not as a sample from a population.\r\n" );
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document.write( "He gave:\r\n" );
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document.write( "4.5^2=20.25\r\n" );
document.write( "20.25*22=445.5 sum of (each data point minus mean)^2\r\n" );
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document.write( "But that is the sum of each data point minus the mean of the 23)^2\r\n" );
document.write( "not (the sum of each data point minus the mean of the 25)^2\r\n" );
document.write( "He incorrectly assumed this was the latter.\r\n" );
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document.write( "It's a bit more complicated than what he did.\r\n" );
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document.write( "He used the definition of the standard deviation\r\n" );
document.write( "of a sample\r\n" );
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document.write( "but you must use the \"shortcut\" formula for the\r\n" );
document.write( "standard deviation of a population \r\n" );
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document.write( "for the standard deviation instead.\r\n" );
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document.write( "Here is the entire correct problem:\r\n" );
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\n" ); document.write( "A certain set of data has a mean of 20 and a Standard deviation of 4.5. It is found that two pieces of data have been omitted, one of value 30 and the other of value 25. Find the mean and S.D of the full 25 pieces of data.
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document.write( "Let  = the unknown sum of the 23 numbers\r\n" );
document.write( "Let  = the unknown sum of the 25 numbers\r\n" );
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document.write( "Then \r\n" );
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document.write( "(1)   \r\n" );
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document.write( "Let  be the given mean of the 23 numbers\r\n" );
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document.write( "Let  be the unknown mean of the 25 numbers. \r\n" );
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document.write( "Since , then\r\n" );
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document.write( "Multiplying both side by 23 gives\r\n" );
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document.write( "From equation (1) above\r\n" );
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document.write( "Therefore\r\n" );
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document.write( "That's the mean of the 25 numbers, which the other tutor \r\n" );
document.write( "calculated correctly.\r\n" );
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document.write( "Now we use the short-cut formula for the standard deviation\r\n" );
document.write( "of a population:\r\n" );
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document.write( "We will use the above formula with both sides squared,\r\n" );
document.write( "for convenience:\r\n" );
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document.write( "Let  be the given standard deviation of the 23 numbers\r\n" );
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document.write( "Let  be the unknown standard deviation of the 25 numbers\r\n" );
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document.write( "Let  be the unknown sum of the squares of the 25 numbers.\r\n" );
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document.write( "Therefore,\r\n" );
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document.write( "(2)  \r\n" );
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document.write( "Using the formula for the 23 numbers:\r\n" );
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document.write( "Substituting given values,\r\n" );
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document.write( "Multiply both sides by 23\r\n" );
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document.write( "From equation (2) above,\r\n" );
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document.write( "Using the formula for the 25 numbers,\r\n" );
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document.write( "Substituting known values\r\n" );
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document.write( "Edwin

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