document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190757>Question 1190757</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The following data give the yearly numbers of law enforcement officers killed in the United <BR>\n" );
document.write( "States over 10 years: 164, 165, 157, 164, 152, 147, 148, 131, 147, 155, calculate sample <BR>\n" );
document.write( "variance and standard deviation </SPAN>\n" );
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document.write( "n = 10 = sample size\r<BR>\n" );
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document.write( "Add up the numbers and you should get 1530<BR>\n" );
document.write( "Divide that over the n = 10 numbers. This yields the sample mean of 1530/10 = 153\r<BR>\n" );
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document.write( "xbar = 153 = sample mean\r<BR>\n" );
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document.write( "Next, subtract the sample mean from each item. <BR>\n" );
document.write( "I recommend making a table to organize the data.<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >x-xbar</TD></tr><tr><TD >164</TD><TD >11</TD></tr><tr><TD >165</TD><TD >12</TD></tr><tr><TD >157</TD><TD >4</TD></tr><tr><TD >164</TD><TD >11</TD></tr><tr><TD >152</TD><TD >-1</TD></tr><tr><TD >147</TD><TD >-6</TD></tr><tr><TD >148</TD><TD >-5</TD></tr><tr><TD >131</TD><TD >-22</TD></tr><tr><TD >147</TD><TD >-6</TD></tr><tr><TD >155</TD><TD >2</TD></tr>\n" );
document.write( "</TABLE>The second column tells us the error of each x value with respect to the center xbar = 153.\r<BR>\n" );
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document.write( "Square each item in the second column to form the third column.<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >x-xbar</TD><TD >(x-xbar)^2</TD></tr><tr><TD >164</TD><TD >11</TD><TD >121</TD></tr><tr><TD >165</TD><TD >12</TD><TD >144</TD></tr><tr><TD >157</TD><TD >4</TD><TD >16</TD></tr><tr><TD >164</TD><TD >11</TD><TD >121</TD></tr><tr><TD >152</TD><TD >-1</TD><TD >1</TD></tr><tr><TD >147</TD><TD >-6</TD><TD >36</TD></tr><tr><TD >148</TD><TD >-5</TD><TD >25</TD></tr><tr><TD >131</TD><TD >-22</TD><TD >484</TD></tr><tr><TD >147</TD><TD >-6</TD><TD >36</TD></tr><tr><TD >155</TD><TD >2</TD><TD >4</TD></tr>\n" );
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document.write( "Add up everything in the third column to get 988. This is the sum of the squared errors (SSE). As the name implies, it's the result of adding up each error squared. Each x-xbar value is the error itself, and we square the item to avoid a negative number. This means any (x-xbar)^2 has a min value of 0.\r<BR>\n" );
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document.write( "Divide this SSE value over n-1 = 10-1 = 9 to compute the sample variance.<BR>\n" );
document.write( "SSE/(n-1) = 988/9 = 109.7778 approximately\r<BR>\n" );
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document.write( "Apply the square root to the sample variance to get the sample standard deviation<BR>\n" );
document.write( "sqrt(988/9) = 10.4775 approximately\r<BR>\n" );
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document.write( "You can use a graphing calculator to check the answers. <BR>\n" );
document.write( "Or you can use a free dedicated online calculator such as this one<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.calculatorsoup.com/calculators/statistics/variance-calculator.php\">https://www.calculatorsoup.com/calculators/statistics/variance-calculator.php</a><BR>\n" );
document.write( "Make sure the \"sample\" option is selected.\r<BR>\n" );
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document.write( "In summary:<BR>\n" );
document.write( "sample variance = 109.7778<BR>\n" );
document.write( "sample standard deviation = 10.4775 <BR>\n" );
document.write( "both values are approximate<BR>\n" );
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