document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171480>Question 1171480</A>: <I><SPAN STYLE=\"word-break: break-all;\"> x    Freq<BR>\n" );
document.write( "1   4<BR>\n" );
document.write( "2   2<BR>\n" );
document.write( "3   4<BR>\n" );
document.write( "6   10<BR>\n" );
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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 #850907 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=850907\">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):** * (1 * 4) + (2 * 2) + (3 * 4) + (6 * 10) = 4 + 4 + 12 + 60 = 80<BR>\n" );
document.write( "* **Total Frequency (n):** * 4 + 2 + 4 + 10 = 20<BR>\n" );
document.write( "* **Mean (x̄):** * 80 / 20 = 4\r<BR>\n" );
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document.write( "**2. Median:**\r<BR>\n" );
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document.write( "* Since there are 20 data points, the median is the average of the 10th and 11th values.<BR>\n" );
document.write( "* Let's list the values in order:<BR>\n" );
document.write( "    * 1 (4 times)<BR>\n" );
document.write( "    * 2 (2 times)<BR>\n" );
document.write( "    * 3 (4 times)<BR>\n" );
document.write( "    * 6 (10 times)<BR>\n" );
document.write( "* The 10th and 11th values are both 6.<BR>\n" );
document.write( "* Therefore, the median is 6.\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( "* 6 appears 10 times, which is the highest frequency.<BR>\n" );
document.write( "* Therefore, the mode is 6.\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( "    * (1 - 4)² * 4 = (-3)² * 4 = 9 * 4 = 36<BR>\n" );
document.write( "    * (2 - 4)² * 2 = (-2)² * 2 = 4 * 2 = 8<BR>\n" );
document.write( "    * (3 - 4)² * 4 = (-1)² * 4 = 1 * 4 = 4<BR>\n" );
document.write( "    * (6 - 4)² * 10 = (2)² * 10 = 4 * 10 = 40<BR>\n" );
document.write( "* **Sum of (x - x̄)² * Freq:**<BR>\n" );
document.write( "    * 36 + 8 + 4 + 40 = 88<BR>\n" );
document.write( "* **Sample Variance (s²):**<BR>\n" );
document.write( "    * 88 / (20 - 1) = 88 / 19 ≈ 4.63\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 = √4.63 ≈ 2.15\r<BR>\n" );
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document.write( "**Summary:**\r<BR>\n" );
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document.write( "* Mean = 4<BR>\n" );
document.write( "* Median = 6<BR>\n" );
document.write( "* Mode = 6<BR>\n" );
document.write( "* Sample Variance ≈ 4.63<BR>\n" );
document.write( "* Sample Standard Deviation ≈ 2.15 </SPAN>\n" );
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