document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166050>Question 1166050</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Question#1<BR>\n" );
document.write( "Following is the sales (in thousands) of salesmen, calculate mean and standard deviation and discuss which salesman performed consistently.<BR>\n" );
document.write( "Salesman1	4	6	5	7	8	11	6	8	10<BR>\n" );
document.write( "Salesman2	26	1	13	1	10	30	4	0	1\r<BR>\n" );
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document.write( "Question#2<BR>\n" );
document.write( "Calculate mean, median and mode of following data.<BR>\n" );
document.write( "Class Interval	1 to 10	11 to 20	21 to 30	31 to 40	41 to 50<BR>\n" );
document.write( "Frequency		6	8	12	14<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 #852870 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</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=852870\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import numpy as np<BR>\n" );
document.write( "import pandas as pd\r<BR>\n" );
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document.write( "# Question 1 Data (Sales in thousands)<BR>\n" );
document.write( "salesman1 = np.array([4, 6, 5, 7, 8, 11, 6, 8, 10])<BR>\n" );
document.write( "salesman2 = np.array([26, 1, 13, 1, 10, 30, 4, 0, 1])\r<BR>\n" );
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document.write( "# Calculate Mean (average)<BR>\n" );
document.write( "mean1 = np.mean(salesman1)<BR>\n" );
document.write( "mean2 = np.mean(salesman2)\r<BR>\n" );
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document.write( "# Calculate Standard Deviation (using sample standard deviation, ddof=1)<BR>\n" );
document.write( "std1 = np.std(salesman1, ddof=1)<BR>\n" );
document.write( "std2 = np.std(salesman2, ddof=1)\r<BR>\n" );
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document.write( "print(f\"Salesman 1 Mean: {mean1}\")<BR>\n" );
document.write( "print(f\"Salesman 1 Standard Deviation: {std1}\")<BR>\n" );
document.write( "print(f\"Salesman 2 Mean: {mean2}\")<BR>\n" );
document.write( "print(f\"Salesman 2 Standard Deviation: {std2}\")\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Salesman 1 Mean: 7.222222222222222<BR>\n" );
document.write( "Salesman 1 Standard Deviation: 2.279132388529557<BR>\n" );
document.write( "Salesman 2 Mean: 9.555555555555555<BR>\n" );
document.write( "Salesman 2 Standard Deviation: 11.413929112175953\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "## Question \#1: Sales Consistency\r<BR>\n" );
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document.write( "To assess consistency, we calculate the **mean** and **standard deviation ($s$)** for each salesman. The standard deviation measures the dispersion of the sales data around the mean; a **lower standard deviation indicates greater consistency**.\r<BR>\n" );
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document.write( "| Statistic | Salesman 1 | Salesman 2 |<BR>\n" );
document.write( "| :---: | :---: | :---: |<BR>\n" );
document.write( "| **Mean ($\bar{X}$)** | $7.22$ (thousand) | $9.56$ (thousand) |<BR>\n" );
document.write( "| **Standard Deviation ($s$)** | $\mathbf{2.28}$ (thousand) | $\mathbf{11.41}$ (thousand) |\r<BR>\n" );
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document.write( "-----\r<BR>\n" );
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document.write( "### Discussion on Consistency\r<BR>\n" );
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document.write( "1.  **Salesman 1** has a mean sales of **$7,220** with a standard deviation of **$2,280**.<BR>\n" );
document.write( "2.  **Salesman 2** has a higher mean sales of **$9,560** but a much larger standard deviation of **$11,410**.\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "**Salesman 1 performed more consistently**. Although Salesman 2 achieved a higher average sale, the high standard deviation indicates that their sales were highly volatile, ranging from 0 to 30 thousand. Salesman 1's sales were tightly clustered around their mean, showing a more predictable and stable performance.\r<BR>\n" );
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document.write( "-----\r<BR>\n" );
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document.write( "## Question \#2: Mean, Median, and Mode for Grouped Data\r<BR>\n" );
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document.write( "The following calculations are based on the four class intervals for which frequency data is provided.\r<BR>\n" );
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document.write( "| Class Interval | Frequency ($f$) | Midpoint ($x_m$) | $f \cdot x_m$ | Cumulative Frequency ($C$) |<BR>\n" );
document.write( "| :---: | :---: | :---: | :---: | :---: |<BR>\n" );
document.write( "| 1 to 10 | 6 | 5.5 | 33.0 | 6 |<BR>\n" );
document.write( "| 11 to 20 | 8 | 15.5 | 124.0 | 14 |<BR>\n" );
document.write( "| 21 to 30 | 12 | 25.5 | 306.0 | 26 |<BR>\n" );
document.write( "| 31 to 40 | 14 | 35.5 | 497.0 | 40 |<BR>\n" );
document.write( "| **Totals** | $\mathbf{N=40}$ | | $\mathbf{\sum f x_m = 960.0}$ | |<BR>\n" );
document.write( "$$\text{Class Width } (h) = 10 \text{ (e.g., } 10 - 1 + 1 \text{ or } 10.5 - 0.5 \text{)}$$\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 $\bar{X} = \frac{\sum f x_m}{N}$.<BR>\n" );
document.write( "$$\bar{X} = \frac{960.0}{40} = \mathbf{24.0}$$\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 splits the data in half. $N/2 = 40/2 = 20$.<BR>\n" );
document.write( "The **median class** is the class where $C$ first exceeds 20, which is **21 to 30** ($C=26$).\r<BR>\n" );
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document.write( "$$\text{Median Formula: } M = L + \left(\frac{\frac{N}{2} - C}{f}\right) \times h$$<BR>\n" );
document.write( "Where:\r<BR>\n" );
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document.write( "  * $L$ (Lower boundary of median class) = 20.5<BR>\n" );
document.write( "  * $N/2 = 20$<BR>\n" );
document.write( "  * $C$ (Cumulative frequency of preceding class) = 14<BR>\n" );
document.write( "  * $f$ (Frequency of median class) = 12<BR>\n" );
document.write( "  * $h$ (Class width) = 10\r<BR>\n" );
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document.write( "$$M = 20.5 + \left(\frac{20 - 14}{12}\right) \times 10$$<BR>\n" );
document.write( "$$M = 20.5 + \left(\frac{6}{12}\right) \times 10$$<BR>\n" );
document.write( "$$M = 20.5 + 5 = \mathbf{25.5}$$\r<BR>\n" );
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document.write( "### 3\. Mode ($Mo$)\r<BR>\n" );
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document.write( "The **modal class** is the class with the highest frequency, which is **31 to 40** ($f_m=14$).\r<BR>\n" );
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document.write( "$$\text{Mode Formula: } Mo = L + \left(\frac{f_m - f_1}{2f_m - f_1 - f_2}\right) \times h$$<BR>\n" );
document.write( "Where:\r<BR>\n" );
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document.write( "  * $L$ (Lower boundary of modal class) = 30.5<BR>\n" );
document.write( "  * $f_m$ (Frequency of modal class) = 14<BR>\n" );
document.write( "  * $f_1$ (Frequency of preceding class) = 12<BR>\n" );
document.write( "  * $f_2$ (Frequency of succeeding class) = 0 (Assuming $f$ for 41-50 is 0)<BR>\n" );
document.write( "  * $h$ (Class width) = 10\r<BR>\n" );
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document.write( "$$Mo = 30.5 + \left(\frac{14 - 12}{2(14) - 12 - 0}\right) \times 10$$<BR>\n" );
document.write( "$$Mo = 30.5 + \left(\frac{2}{28 - 12}\right) \times 10$$<BR>\n" );
document.write( "$$Mo = 30.5 + \left(\frac{2}{16}\right) \times 10$$<BR>\n" );
document.write( "$$Mo = 30.5 + 1.25 = \mathbf{31.75}$$ </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );