document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178024>Question 1178024</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The final grade in statistics of 80 student at AAMUSTED Mathematics level100 are recorded below,<BR>\n" );
document.write( "68 84 75 82 68 90 62 88 76 93<BR>\n" );
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document.write( "73 79 88 73 60 93 71 59 85 75\r<BR>\n" );
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document.write( "61 65 75 87 74 62 95 78 63 72\r<BR>\n" );
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document.write( "66 78 82 75 94 77 69 74 68 60\r<BR>\n" );
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document.write( "96 78 89 61 75 95 60 79 83 71\r<BR>\n" );
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document.write( "79 62 67 97 78 85 76 65 71 75\r<BR>\n" );
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document.write( "65 80 73 57 88 78 62 76 53 74\r<BR>\n" );
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document.write( "86 67 73 81 72 63 76 75 85 77\r<BR>\n" );
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document.write( "1:Using the sturge's approximation rule construct a frequency distribution table for the data above<BR>\n" );
document.write( "2:Use the table to calculate;<BR>\n" );
document.write( "* Mean and standard deviation<BR>\n" );
document.write( "* Skewness and kurtosis.\r<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 #850386 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=850386\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Sturge's Approximation Rule**\r<BR>\n" );
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document.write( "Sturge's rule helps determine the number of classes (k) for a frequency distribution:\r<BR>\n" );
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document.write( "* k = 1 + 3.322 * log10(n)\r<BR>\n" );
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document.write( "Where n is the number of data points (n = 80 in this case).\r<BR>\n" );
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document.write( "* k = 1 + 3.322 * log10(80)<BR>\n" );
document.write( "* k = 1 + 3.322 * 1.9031<BR>\n" );
document.write( "* k ≈ 1 + 6.322<BR>\n" );
document.write( "* k ≈ 7.322\r<BR>\n" );
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document.write( "We round k to the nearest whole number, so k = 7.\r<BR>\n" );
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document.write( "**2. Range and Class Width**\r<BR>\n" );
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document.write( "* **Minimum Value:** 53<BR>\n" );
document.write( "* **Maximum Value:** 97<BR>\n" );
document.write( "* **Range:** 97 - 53 = 44\r<BR>\n" );
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document.write( "* **Class Width (w):** Range / k = 44 / 7 ≈ 6.286\r<BR>\n" );
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document.write( "We round the class width up to the nearest convenient whole number, so w = 7.\r<BR>\n" );
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document.write( "**3. Frequency Distribution Table**\r<BR>\n" );
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document.write( "| Class Interval | Class Midpoint (x) | Frequency (f) | fx | f(x-mean)^2 |<BR>\n" );
document.write( "|----------------|--------------------|---------------|----|-------------|<BR>\n" );
document.write( "| 53 - 59        | 56                 | 3             | 168 | 2755.07     |<BR>\n" );
document.write( "| 60 - 66        | 63                 | 11            | 693 | 1968.64     |<BR>\n" );
document.write( "| 67 - 73        | 70                 | 13            | 910 | 258.91      |<BR>\n" );
document.write( "| 74 - 80        | 77                 | 22            | 1694 | 2.64        |<BR>\n" );
document.write( "| 81 - 87        | 84                 | 8             | 672 | 1146.64     |<BR>\n" );
document.write( "| 88 - 94        | 91                 | 12            | 1092 | 2673.91     |<BR>\n" );
document.write( "| 95 - 101       | 98                 | 11            | 1078 | 4991.64     |<BR>\n" );
document.write( "| **Total** |                    | **80** | **6307** | **13828.05** |\r<BR>\n" );
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document.write( "**4. Calculations**\r<BR>\n" );
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document.write( "* **Mean (x̄):** Σfx / n = 6307 / 80 ≈ 78.8375\r<BR>\n" );
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document.write( "* **Standard Deviation (s):**<BR>\n" );
document.write( "    * s = √[Σf(x - x̄)² / (n - 1)]<BR>\n" );
document.write( "    * s = √[13828.05 / 79]<BR>\n" );
document.write( "    * s = √175.0386 ≈ 13.23\r<BR>\n" );
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document.write( "**5. Skewness and Kurtosis**\r<BR>\n" );
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document.write( "For this, we'll need to calculate the third and fourth moments.\r<BR>\n" );
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document.write( "* **Third Moment (m3):** Σf(x - x̄)³ / n<BR>\n" );
document.write( "* **Fourth Moment (m4):** Σf(x - x̄)⁴ / n\r<BR>\n" );
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document.write( "Let's add those columns to our table.\r<BR>\n" );
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document.write( "| Class Interval | Class Midpoint (x) | Frequency (f) | fx | f(x-mean)^2 | f(x-mean)^3 | f(x-mean)^4 |<BR>\n" );
document.write( "|----------------|--------------------|---------------|----|-------------|-------------|-------------|<BR>\n" );
document.write( "| 53 - 59        | 56                 | 3             | 168 | 2755.07     | -13636.57    | 674482.16   |<BR>\n" );
document.write( "| 60 - 66        | 63                 | 11            | 693 | 1968.64     | -6922.82     | 243288.58   |<BR>\n" );
document.write( "| 67 - 73        | 70                 | 13            | 910 | 258.91      | -414.07      | 6625.16     |<BR>\n" );
document.write( "| 74 - 80        | 77                 | 22            | 1694 | 2.64        | -0.82        | 0.25        |<BR>\n" );
document.write( "| 81 - 87        | 84                 | 8             | 672 | 1146.64     | 3833.18      | 128362.43   |<BR>\n" );
document.write( "| 88 - 94        | 91                 | 12            | 1092 | 2673.91     | 13783.50     | 710892.05   |<BR>\n" );
document.write( "| 95 - 101       | 98                 | 11            | 1078 | 4991.64     | 33504.60     | 2252110.82  |<BR>\n" );
document.write( "| **Total** |                    | **80** | **6307** | **13828.05** | **40567.00** | **3336361.45** |\r<BR>\n" );
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document.write( "* **m3:** 40567 / 80 ≈ 507.0875<BR>\n" );
document.write( "* **m4:** 3336361.45 / 80 ≈ 41704.5181\r<BR>\n" );
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document.write( "* **Skewness (g1):** m3 / s³ = 507.0875 / 13.23³ ≈ 507.0875 / 2315.68 ≈ 0.219<BR>\n" );
document.write( "* **Kurtosis (g2):** m4 / s⁴ - 3 = 41704.5181 / 13.23⁴ - 3 ≈ 41704.5181 / 30638.15 - 3 ≈ 1.361 - 3 ≈ -1.639\r<BR>\n" );
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document.write( "**Results**\r<BR>\n" );
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document.write( "1.  **Frequency Distribution Table:** As shown above.<BR>\n" );
document.write( "2.  **Calculations:**<BR>\n" );
document.write( "    * **Mean:** ≈ 78.84<BR>\n" );
document.write( "    * **Standard Deviation:** ≈ 13.23<BR>\n" );
document.write( "    * **Skewness:** ≈ 0.219 (slightly positive skew)<BR>\n" );
document.write( "    * **Kurtosis:** ≈ -1.639 (platykurtic, flatter than normal)<BR>\n" );
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