document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166623>Question 1166623</A>: <I><SPAN STYLE=\"word-break: break-all;\">  The weights (in kg) of 35 persons are given below:<BR>\n" );
document.write( "43, 51, 47, 62, 48, 40, 50, 62, 53, 56, 40, 48, 56, 53, 50, 42, 55, 52, 48, 46, 45, 54, 52, 50, 47, 44, 54, 55, 60, 63, 58, 55, 60, 58, 53<BR>\n" );
document.write( "a) Group this data by using the Sturges’ formula to determine the number of classes. Keep the width of each class equal.<BR>\n" );
document.write( "b) Find the mean, median, standard deviation and variance of the grouped data obtained in (a) above. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852740 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.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=852740\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a great exercise in statistical data grouping and analysis. Here are the steps to solve the problem.\r<BR>\n" );
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document.write( "## 📊 a) Grouping the Data\r<BR>\n" );
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document.write( "### 1. Find the Range ($R$)\r<BR>\n" );
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document.write( "The range is the difference between the highest and lowest values in the data set.\r<BR>\n" );
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document.write( "* **Highest Value ($H$):** 63 kg<BR>\n" );
document.write( "* **Lowest Value ($L$):** 40 kg<BR>\n" );
document.write( "* $$R = H - L = 63 - 40 = 23$$\r<BR>\n" );
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document.write( "### 2. Determine the Number of Classes ($k$) using Sturges' Formula\r<BR>\n" );
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document.write( "Sturges' formula is used to estimate the optimal number of class intervals.<BR>\n" );
document.write( "$$k = 1 + 3.322 \log_{10}(n)$$<BR>\n" );
document.write( "where $n$ is the total number of observations ($n = 35$).\r<BR>\n" );
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document.write( "$$k = 1 + 3.322 \log_{10}(35)$$<BR>\n" );
document.write( "$$k \approx 1 + 3.322(1.544)$$<BR>\n" );
document.write( "$$k \approx 1 + 5.132$$<BR>\n" );
document.write( "$$k \approx 6.132$$\r<BR>\n" );
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document.write( "We round this number up to the nearest integer to get the number of classes.<BR>\n" );
document.write( "$$\mathbf{k = 7 \text{ classes}}$$\r<BR>\n" );
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document.write( "### 3. Determine the Class Width ($w$)\r<BR>\n" );
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document.write( "The class width is the range divided by the number of classes.\r<BR>\n" );
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document.write( "$$w = \frac{R}{k} = \frac{23}{7} \approx 3.286$$\r<BR>\n" );
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document.write( "We must choose a convenient, round number slightly greater than 3.286 to ensure all data points are covered. Let's choose a class width of $\mathbf{4}$.\r<BR>\n" );
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document.write( "### 4. Create the Frequency Distribution Table\r<BR>\n" );
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document.write( "Starting the first class boundary at the lowest value, 40, and using a width of 4:\r<BR>\n" );
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document.write( "| Class Interval (Weight in kg) | Tally | Frequency ($f$) |<BR>\n" );
document.write( "| :---: | :---: | :---: |<BR>\n" );
document.write( "| 40 - 43 | IIII | 4 |<BR>\n" );
document.write( "| 44 - 47 | IIII I | 6 |<BR>\n" );
document.write( "| 48 - 51 | IIII IIII | 9 |<BR>\n" );
document.write( "| 52 - 55 | IIII IIII | 9 |<BR>\n" );
document.write( "| 56 - 59 | IIII | 5 |<BR>\n" );
document.write( "| 60 - 63 | II | 2 |<BR>\n" );
document.write( "| **Total** | | **35** |\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 💻 b) Find Mean, Median, Standard Deviation, and Variance\r<BR>\n" );
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document.write( "To calculate these statistics for **grouped data**, we must use the **class midpoint ($x$)** to represent the values in each interval.\r<BR>\n" );
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document.write( "| Class Interval | Midpoint ($x$) | Frequency ($f$) | $f \cdot x$ | $|x - \bar{x}|$ | $f \cdot |x - \bar{x}|$ | $x - \bar{x}$ | $(x - \bar{x})^2$ | $f \cdot (x - \bar{x})^2$ |<BR>\n" );
document.write( "| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |<BR>\n" );
document.write( "| 40 - 43 | 41.5 | 4 | 166.0 | 9.07 | 36.28 | -9.07 | 82.26 | 329.04 |<BR>\n" );
document.write( "| 44 - 47 | 45.5 | 6 | 273.0 | 5.07 | 30.42 | -5.07 | 25.70 | 154.20 |<BR>\n" );
document.write( "| 48 - 51 | 49.5 | 9 | 445.5 | 1.07 | 9.63 | -1.07 | 1.14 | 10.26 |<BR>\n" );
document.write( "| 52 - 55 | 53.5 | 9 | 481.5 | 2.93 | 26.37 | 2.93 | 8.58 | 77.22 |<BR>\n" );
document.write( "| 56 - 59 | 57.5 | 5 | 287.5 | 6.93 | 34.65 | 6.93 | 47.98 | 239.90 |<BR>\n" );
document.write( "| 60 - 63 | 61.5 | 2 | 123.0 | 10.93 | 21.86 | 10.93 | 119.46 | 238.92 |<BR>\n" );
document.write( "| **Total** | | $\sum f = 35$ | $\sum fx = 1776.5$ | | | | | $\sum f(x - \bar{x})^2 = 1049.54$ |\r<BR>\n" );
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document.write( "### 1. Mean ($\bar{x}$)\r<BR>\n" );
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document.write( "The mean for grouped data is calculated as:<BR>\n" );
document.write( "$$\bar{x} = \frac{\sum f x}{\sum f}$$<BR>\n" );
document.write( "$$\bar{x} = \frac{1776.5}{35} \approx \mathbf{50.76 \text{ kg}}$$\r<BR>\n" );
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document.write( "### 2. Median ($M$)\r<BR>\n" );
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document.write( "The median is the value that falls at the $n/2$ position.<BR>\n" );
document.write( "$$\frac{n}{2} = \frac{35}{2} = 17.5$$<BR>\n" );
document.write( "The median class is the first class whose cumulative frequency is $\geq 17.5$.<BR>\n" );
document.write( "* Cumulative Frequencies: 4, 10, **19**, 28, 33, 35.<BR>\n" );
document.write( "* The median class is **48 - 51** (Cumulative Frequency 19).\r<BR>\n" );
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document.write( "Median Formula:<BR>\n" );
document.write( "$$M = L + \left(\frac{\frac{n}{2} - C_f}{f_m}\right) \cdot w$$<BR>\n" );
document.write( "Where:<BR>\n" );
document.write( "* $L$: Lower boundary of the median class ($48 - 0.5 = 47.5$)<BR>\n" );
document.write( "* $n$: Total frequency (35)<BR>\n" );
document.write( "* $C_f$: Cumulative frequency of the class before the median class (10)<BR>\n" );
document.write( "* $f_m$: Frequency of the median class (9)<BR>\n" );
document.write( "* $w$: Class width (4)\r<BR>\n" );
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document.write( "$$M = 47.5 + \left(\frac{17.5 - 10}{9}\right) \cdot 4$$<BR>\n" );
document.write( "$$M = 47.5 + \left(\frac{7.5}{9}\right) \cdot 4$$<BR>\n" );
document.write( "$$M = 47.5 + 0.833 \cdot 4$$<BR>\n" );
document.write( "$$M = 47.5 + 3.332 \approx \mathbf{50.83 \text{ kg}}$$\r<BR>\n" );
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document.write( "### 3. Variance ($\sigma^2$)\r<BR>\n" );
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document.write( "The variance is calculated using the formula:<BR>\n" );
document.write( "$$\sigma^2 = \frac{\sum f(x - \bar{x})^2}{n - 1}$$<BR>\n" );
document.write( "Using $n=35$:<BR>\n" );
document.write( "$$\sigma^2 = \frac{1049.54}{35 - 1} = \frac{1049.54}{34} \approx \mathbf{30.87}$$\r<BR>\n" );
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document.write( "### 4. Standard Deviation ($\sigma$)\r<BR>\n" );
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document.write( "The standard deviation is the square root of the variance:<BR>\n" );
document.write( "$$\sigma = \sqrt{\sigma^2}$$<BR>\n" );
document.write( "$$\sigma = \sqrt{30.87} \approx \mathbf{5.56 \text{ kg}}$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 📝 Summary of Results\r<BR>\n" );
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document.write( "| Statistic | Value (Grouped Data) |<BR>\n" );
document.write( "| :---: | :---: |<BR>\n" );
document.write( "| **Mean** ($\bar{x}$) | $50.76$ kg |<BR>\n" );
document.write( "| **Median** ($M$) | $50.83$ kg |<BR>\n" );
document.write( "| **Variance** ($\sigma^2$) | $30.87$ |<BR>\n" );
document.write( "| **Standard Deviation** ($\sigma$) | $5.56$ kg | </SPAN>\n" );
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