document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171481>Question 1171481</A>: <I><SPAN STYLE=\"word-break: break-all;\">   x     Freq<BR>\n" );
document.write( "100   11<BR>\n" );
document.write( "90      2<BR>\n" );
document.write( "70      6<BR>\n" );
document.write( "50      6<BR>\n" );
document.write( "mean = <BR>\n" );
document.write( "median = <BR>\n" );
document.write( "mode = <BR>\n" );
document.write( "sample variance = <BR>\n" );
document.write( "sample standard deviation =  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850906 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850906\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's calculate the mean, median, mode, sample variance, and sample standard deviation for the given data.\r<BR>\n" );
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document.write( "**1. Mean:**\r<BR>\n" );
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document.write( "* **Sum of (x * Freq):**<BR>\n" );
document.write( "    * (100 * 11) + (90 * 2) + (70 * 6) + (50 * 6) = 1100 + 180 + 420 + 300 = 2000<BR>\n" );
document.write( "* **Total Frequency (n):**<BR>\n" );
document.write( "    * 11 + 2 + 6 + 6 = 25<BR>\n" );
document.write( "* **Mean (x̄):**<BR>\n" );
document.write( "    * 2000 / 25 = 80\r<BR>\n" );
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document.write( "**2. Median:**\r<BR>\n" );
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document.write( "* Since there are 25 data points, the median is the (25 + 1) / 2 = 13th value.<BR>\n" );
document.write( "* Let's list the values in order:<BR>\n" );
document.write( "    * 100 (11 times)<BR>\n" );
document.write( "    * 90 (2 times)<BR>\n" );
document.write( "    * 70 (6 times)<BR>\n" );
document.write( "    * 50 (6 times)<BR>\n" );
document.write( "* The 13th value falls within the 70's.<BR>\n" );
document.write( "* Therefore, the median is 70.\r<BR>\n" );
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document.write( "**3. Mode:**\r<BR>\n" );
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document.write( "* The mode is the value that appears most frequently.<BR>\n" );
document.write( "* 100 appears 11 times, which is the highest frequency.<BR>\n" );
document.write( "* Therefore, the mode is 100.\r<BR>\n" );
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document.write( "**4. Sample Variance (s²):**\r<BR>\n" );
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document.write( "* **Calculate (x - x̄)² * Freq:**<BR>\n" );
document.write( "    * (100 - 80)² * 11 = 20² * 11 = 400 * 11 = 4400<BR>\n" );
document.write( "    * (90 - 80)² * 2 = 10² * 2 = 100 * 2 = 200<BR>\n" );
document.write( "    * (70 - 80)² * 6 = (-10)² * 6 = 100 * 6 = 600<BR>\n" );
document.write( "    * (50 - 80)² * 6 = (-30)² * 6 = 900 * 6 = 5400<BR>\n" );
document.write( "* **Sum of (x - x̄)² * Freq:**<BR>\n" );
document.write( "    * 4400 + 200 + 600 + 5400 = 10600<BR>\n" );
document.write( "* **Sample Variance (s²):**<BR>\n" );
document.write( "    * 10600 / (25 - 1) = 10600 / 24 ≈ 441.67\r<BR>\n" );
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document.write( "**5. Sample Standard Deviation (s):**\r<BR>\n" );
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document.write( "* **Sample Standard Deviation (s):**<BR>\n" );
document.write( "    * √s² = √441.67 ≈ 21.02\r<BR>\n" );
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document.write( "**Summary:**\r<BR>\n" );
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document.write( "* Mean = 80<BR>\n" );
document.write( "* Median = 70<BR>\n" );
document.write( "* Mode = 100<BR>\n" );
document.write( "* Sample Variance ≈ 441.67<BR>\n" );
document.write( "* Sample Standard Deviation ≈ 21.02<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );