document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191696>Question 1191696</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2.	Then grade point averages of 20 college students selected at random from the graduating class are as follows:\r<BR>\n" );
document.write( "\n" );
document.write( "3.2	1.9	2.7	2.4<BR>\n" );
document.write( "2.8	2.9	3.8	3.0<BR>\n" );
document.write( "			2.5	3.3	1.8	2.5<BR>\n" );
document.write( "			3.7	2.8	2.0	3.2<BR>\n" );
document.write( "			2.3	2.1	2.5	1.9\r<BR>\n" );
document.write( "\n" );
document.write( "a.	Calculate the descriptive statistics.<BR>\n" );
document.write( "b.	Use the appropriate graph for the GPA of the 20 students. <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849145 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=849145\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's the solution:\r<BR>\n" );
document.write( "\n" );
document.write( "**a. Descriptive Statistics**\r<BR>\n" );
document.write( "\n" );
document.write( "First, let's organize the data in ascending order to make calculations easier:\r<BR>\n" );
document.write( "\n" );
document.write( "1.8, 1.9, 1.9, 2.0, 2.1, 2.3, 2.4, 2.5, 2.5, 2.5, 2.7, 2.8, 2.8, 2.9, 3.0, 3.2, 3.2, 3.3, 3.7, 3.8\r<BR>\n" );
document.write( "\n" );
document.write( "Now we can calculate the descriptive statistics:\r<BR>\n" );
document.write( "\n" );
document.write( "* **Mean:** (Sum of all values) / (Number of values)<BR>\n" );
document.write( "   Mean = (1.8 + 1.9 + 1.9 + 2.0 + 2.1 + 2.3 + 2.4 + 2.5 + 2.5 + 2.5 + 2.7 + 2.8 + 2.8 + 2.9 + 3.0 + 3.2 + 3.2 + 3.3 + 3.7 + 3.8) / 20<BR>\n" );
document.write( "   Mean = 51.5 / 20 = 2.575\r<BR>\n" );
document.write( "\n" );
document.write( "* **Median:** The middle value (since there are 20 values, the median is the average of the 10th and 11th values)<BR>\n" );
document.write( "   Median = (2.5 + 2.7) / 2 = 2.6\r<BR>\n" );
document.write( "\n" );
document.write( "* **Mode:** The most frequent value(s)<BR>\n" );
document.write( "   Mode = 2.5 (appears 3 times)\r<BR>\n" );
document.write( "\n" );
document.write( "* **Range:** Maximum value - Minimum value<BR>\n" );
document.write( "   Range = 3.8 - 1.8 = 2.0\r<BR>\n" );
document.write( "\n" );
document.write( "* **Variance:**  The average of the squared differences from the mean.  Here's how to calculate it:<BR>\n" );
document.write( "    1. Subtract the mean from each value and square the result.<BR>\n" );
document.write( "    2. Sum all the squared differences.<BR>\n" );
document.write( "    3. Divide the sum by the number of values minus 1 (n-1 = 20-1 =19 for sample variance). This is called the sample variance.<BR>\n" );
document.write( "    Variance ≈ 0.355 (You can use a calculator or spreadsheet software for this).\r<BR>\n" );
document.write( "\n" );
document.write( "* **Standard Deviation:** The square root of the variance<BR>\n" );
document.write( "    Standard Deviation ≈ √0.355 ≈ 0.596\r<BR>\n" );
document.write( "\n" );
document.write( "**b. Appropriate Graph**\r<BR>\n" );
document.write( "\n" );
document.write( "A **histogram** or a **stem-and-leaf plot** would be appropriate for visualizing the GPA data.  A histogram is generally preferred for larger datasets.  Here's what a histogram would look like conceptually:\r<BR>\n" );
document.write( "\n" );
document.write( "* **X-axis:** GPA ranges (e.g., 1.8-2.0, 2.0-2.2, 2.2-2.4, ..., 3.8-4.0)<BR>\n" );
document.write( "* **Y-axis:** Frequency (number of students in each GPA range)\r<BR>\n" );
document.write( "\n" );
document.write( "You would then create bars for each GPA range, with the height of the bar representing the number of students whose GPAs fall within that range.\r<BR>\n" );
document.write( "\n" );
document.write( "A stem-and-leaf plot would also be suitable, especially since the dataset is relatively small.  It would show the individual data points while still giving a sense of the distribution.  Here's how it would look:\r<BR>\n" );
document.write( "\n" );
document.write( "```<BR>\n" );
document.write( "1 | 899<BR>\n" );
document.write( "2 | 01345557889<BR>\n" );
document.write( "3 | 022378<BR>\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "Where the \"stem\" is the digit before the decimal point, and the \"leaves\" are the digits after the decimal point.\r<BR>\n" );
document.write( "\n" );
document.write( "Both the histogram and stem-and-leaf plot would give you a visual representation of the distribution of GPAs among the 20 students.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );