document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1119164>Question 1119164</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A distribution has a standard deviation of &#417; = 4. Find the z-score for each of the following locations in the distributions.  Explain the logic and show the math processes for each score.<BR>\n" );
document.write( "a. Above the mean by 4 points.<BR>\n" );
document.write( "b. Above the mean by 12 points.<BR>\n" );
document.write( "c. Below the mean by 2 points.<BR>\n" );
document.write( "d. Below the mean by 8 points.\r<BR>\n" );
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document.write( "I cannot stand learning from the text book! If someone can show me a visual breakdown on how to solve the equation that would be fantastic! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #735101 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=735101\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> standard deviation is 4.\r<BR>\n" );
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document.write( "z-score is equal to (x-m)/s.\r<BR>\n" );
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document.write( "x is the raw score.<BR>\n" );
document.write( "m is the mean.<BR>\n" );
document.write( "s is the standard deviation.\r<BR>\n" );
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document.write( "when you're dealing with a z-score, the mean is equal to 0 and the standard deviation is equal to 1.\r<BR>\n" );
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document.write( "the z-score tells you how many standard deviations you are above or below the mean.\r<BR>\n" );
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document.write( "if your standard deviation is 4 points, then the formula becomes:\r<BR>\n" );
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document.write( "z = (x-m)/4\r<BR>\n" );
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document.write( "if your raw score is 4 points above the mean, then x = m + 4 and the formula becomes:\r<BR>\n" );
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document.write( "z = (m + 4 - m) / 4 which simplifies to z = 4/4 = 1.\r<BR>\n" );
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document.write( "z = 1 means the raw score of 4 points above the mean is 1 standard deviation above the mean.\r<BR>\n" );
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document.write( "if your raw score is 12 points above the mean, then x = m + 12 and the formula becomes:\r<BR>\n" );
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document.write( "z = (m + 12 - m) / 4 which simplifies to z = 12/4 = 3.\r<BR>\n" );
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document.write( "z = 3 means the raw score of 12 points above the mean is 3 standard deviations above the mean.\r<BR>\n" );
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document.write( "when the raw score is 2 points below the mean, then x = m - 2 and the formula becomes:\r<BR>\n" );
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document.write( "z = (m - 2 - m) / 4 which simplifies to z = -2/4 = -1/2.\r<BR>\n" );
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document.write( "z = -1/2 means the raw score of 2 points below the mean is 1/2 standard deviations below the mean.\r<BR>\n" );
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document.write( "when the raw score is 8 points below the mean, then x = m - 8 and the formula becomes:\r<BR>\n" );
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document.write( "z = (m - 8 - m) / 4 which simplifies to z = -2.\r<BR>\n" );
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document.write( "z = -2 means the raw score of 8 points below the mean is 2 standard deviations below the mean.\r<BR>\n" );
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document.write( "note that the z-score of the mean follows the same formula of z = (x-m)/s.\r<BR>\n" );
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document.write( "when x is equal to the mean, the formula becomes z = (m-m)/s, which becomes z = 0.\r<BR>\n" );
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document.write( "the z-score of the mean is always equal to 0.\r<BR>\n" );
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document.write( "the z-score tells you how many standard deviations you are above or below the mean.\r<BR>\n" );
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document.write( "graphically, there's a nice calculator that will show you what this looks like in terms of the normal distribution curve.\r<BR>\n" );
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document.write( "this calculator can be found at <a  rel=nofollow HREF= \"http://davidmlane.com/hyperstat/z_table.html\" target = \"_blank\">http://davidmlane.com/hyperstat/z_table.html</a>\r<BR>\n" );
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document.write( "you can use this calculator with the mean of the raw score and the standard deviation of the raw score distribution.\r<BR>\n" );
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document.write( "you can also use this calculator with the mean of the z-score and the standard deviation of the z-score.\r<BR>\n" );
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document.write( "you still need the formula to translate from the raw score to the z-score and vice versa.\r<BR>\n" );
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document.write( "to confirm the calculations are correct, we can use the area to the left of the z-score and the raw score to confirm they are the same point under the normal distribution curve.\r<BR>\n" );
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document.write( "we can assume any mean for the raw score mean since all the measurements are relative to the mean and the standard deviation is fixed at 4 points.\r<BR>\n" );
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document.write( "we'll assume the raw score mean is 5000.\r<BR>\n" );
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document.write( "we'll do 4 points and 12 points above the mean and 2 points and 8 points below the mean in that order.\r<BR>\n" );
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document.write( "each one will be in pairs with the raw score first and the z-score following.\r<BR>\n" );
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document.write( "the graphs are shown below:\r<BR>\n" );
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document.write( "4 points above the mean:\r<BR>\n" );
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document.write( "12 points above the mean:\r<BR>\n" );
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document.write( "2 points below the mean:\r<BR>\n" );
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document.write( "8 points below the mean:\r<BR>\n" );
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document.write( "you can see that the area to the left of the raw score and the area to the left of the z-score are comparable for each pair.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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