document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1175464>Question 1175464</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the following grouped frequency distribution, which portrays the duration (in minutes) of 50 lectures<BR>\n" );
document.write( "delivered by a Chemistry professor.\r<BR>\n" );
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document.write( "Duration (Minutes) Frequency<BR>\n" );
document.write( "30.00 < 40.00 3<BR>\n" );
document.write( "40.00 < 50.00 7<BR>\n" );
document.write( "50.00 < 60.00 18<BR>\n" );
document.write( "60.00 < 70.00 13<BR>\n" );
document.write( "70.00 < 80.00 8<BR>\n" );
document.write( "80.00 < 90.00 1\r<BR>\n" );
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document.write( "Using computational formulae, calculate the:\r<BR>\n" );
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document.write( "(i) estimated mean. <BR>\n" );
document.write( "(ii) mode. <BR>\n" );
document.write( "(iii) standard deviation. <BR>\n" );
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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 #850616 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=850616\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's calculate the estimated mean, mode, and standard deviation for this grouped frequency distribution.\r<BR>\n" );
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document.write( "**1. Estimated Mean**\r<BR>\n" );
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document.write( "* **Midpoints (x_m):** Find the midpoint of each class interval.<BR>\n" );
document.write( "    * 30-40: 35<BR>\n" );
document.write( "    * 40-50: 45<BR>\n" );
document.write( "    * 50-60: 55<BR>\n" );
document.write( "    * 60-70: 65<BR>\n" );
document.write( "    * 70-80: 75<BR>\n" );
document.write( "    * 80-90: 85<BR>\n" );
document.write( "* **Multiply Midpoints by Frequencies (f * x_m):**<BR>\n" );
document.write( "    * 35 * 3 = 105<BR>\n" );
document.write( "    * 45 * 7 = 315<BR>\n" );
document.write( "    * 55 * 18 = 990<BR>\n" );
document.write( "    * 65 * 13 = 845<BR>\n" );
document.write( "    * 75 * 8 = 600<BR>\n" );
document.write( "    * 85 * 1 = 85<BR>\n" );
document.write( "* **Sum of (f * x_m):** 105 + 315 + 990 + 845 + 600 + 85 = 2940<BR>\n" );
document.write( "* **Sum of Frequencies (n):** 3 + 7 + 18 + 13 + 8 + 1 = 50<BR>\n" );
document.write( "* **Estimated Mean (x̄):** (Σ f * x_m) / n = 2940 / 50 = 58.8\r<BR>\n" );
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document.write( "**(i) Estimated Mean = 58.8 minutes**\r<BR>\n" );
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document.write( "**2. Mode**\r<BR>\n" );
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document.write( "* **Modal Class:** The class with the highest frequency is 50-60 (frequency = 18).<BR>\n" );
document.write( "* **Lower Boundary (L):** 50<BR>\n" );
document.write( "* **Class Width (w):** 10<BR>\n" );
document.write( "* **Frequency of Modal Class (f_m):** 18<BR>\n" );
document.write( "* **Frequency of Class Before Modal Class (f_1):** 7<BR>\n" );
document.write( "* **Frequency of Class After Modal Class (f_2):** 13<BR>\n" );
document.write( "* **Mode Formula:** L + [(f_m - f_1) / ((f_m - f_1) + (f_m - f_2))] * w<BR>\n" );
document.write( "    * Mode = 50 + [(18 - 7) / ((18 - 7) + (18 - 13))] * 10<BR>\n" );
document.write( "    * Mode = 50 + [11 / (11 + 5)] * 10<BR>\n" );
document.write( "    * Mode = 50 + (11 / 16) * 10<BR>\n" );
document.write( "    * Mode = 50 + (0.6875) * 10<BR>\n" );
document.write( "    * Mode = 50 + 6.875<BR>\n" );
document.write( "    * Mode = 56.875\r<BR>\n" );
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document.write( "**(ii) Mode = 56.875 minutes**\r<BR>\n" );
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document.write( "**3. Standard Deviation**\r<BR>\n" );
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document.write( "* **Calculate (x_m - x̄)²:**<BR>\n" );
document.write( "    * (35 - 58.8)² = (-23.8)² = 566.44<BR>\n" );
document.write( "    * (45 - 58.8)² = (-13.8)² = 190.44<BR>\n" );
document.write( "    * (55 - 58.8)² = (-3.8)² = 14.44<BR>\n" );
document.write( "    * (65 - 58.8)² = (6.2)² = 38.44<BR>\n" );
document.write( "    * (75 - 58.8)² = (16.2)² = 262.44<BR>\n" );
document.write( "    * (85 - 58.8)² = (26.2)² = 686.44<BR>\n" );
document.write( "* **Multiply by Frequencies (f * (x_m - x̄)²):**<BR>\n" );
document.write( "    * 3 * 566.44 = 1699.32<BR>\n" );
document.write( "    * 7 * 190.44 = 1333.08<BR>\n" );
document.write( "    * 18 * 14.44 = 259.92<BR>\n" );
document.write( "    * 13 * 38.44 = 499.72<BR>\n" );
document.write( "    * 8 * 262.44 = 2099.52<BR>\n" );
document.write( "    * 1 * 686.44 = 686.44<BR>\n" );
document.write( "* **Sum of (f * (x_m - x̄)²):** 1699.32 + 1333.08 + 259.92 + 499.72 + 2099.52 + 686.44 = 6578.00<BR>\n" );
document.write( "* **Variance (s²):** (Σ f * (x_m - x̄)²) / (n - 1) = 6578.00 / (50 - 1) = 6578 / 49 = 134.2449<BR>\n" );
document.write( "* **Standard Deviation (s):** √Variance = √134.2449 ≈ 11.5864\r<BR>\n" );
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document.write( "**(iii) Standard Deviation ≈ 11.59 minutes**<BR>\n" );
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