document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166221>Question 1166221</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Mr Cheelo�s perfomance of his 170 Imionology class of students is given in an incomplete distribution below.\r<BR>\n" );
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document.write( "Variable      0-10 10-20 20-30 30-40 40-50 50-60 60-70 <BR>\n" );
document.write( "Frequence      10   20     f1   40     f2   25     15\r<BR>\n" );
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document.write( "(a)  If the median is 35.  Find the missing frequence.\r<BR>\n" );
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document.write( "(b)  Calculate mean, Interquartile range and standard deviation.  Hence explain using the mean and standard devi-ation, the meaning of these values in relation to Mr cheelo results. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852863 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=852863\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a problem involving calculations with a **grouped frequency distribution**, specifically finding missing frequencies using the median, and then calculating key statistics like the mean, interquartile range (IQR), and standard deviation.\r<BR>\n" );
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document.write( "The total number of students is $N = 170$.<BR>\n" );
document.write( "The frequency distribution is:\r<BR>\n" );
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document.write( "| Variable (Score) | Frequency ($f$) |<BR>\n" );
document.write( "| :---: | :---: |<BR>\n" );
document.write( "| 0-10 | 10 |<BR>\n" );
document.write( "| 10-20 | 20 |<BR>\n" );
document.write( "| 20-30 | $f_1$ |<BR>\n" );
document.write( "| 30-40 | 40 |<BR>\n" );
document.write( "| 40-50 | $f_2$ |<BR>\n" );
document.write( "| 50-60 | 25 |<BR>\n" );
document.write( "| 60-70 | 15 |\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## (a) Finding the Missing Frequencies\r<BR>\n" );
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document.write( "We have two missing frequencies, $f_1$ and $f_2$. We'll use two pieces of information: the total frequency ($N$) and the median value.\r<BR>\n" );
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document.write( "### 1. Using the Total Frequency ($N=170$)\r<BR>\n" );
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document.write( "The sum of all frequencies must equal the total number of students:<BR>\n" );
document.write( "$$10 + 20 + f_1 + 40 + f_2 + 25 + 15 = 170$$<BR>\n" );
document.write( "$$110 + f_1 + f_2 = 170$$<BR>\n" );
document.write( "$$f_1 + f_2 = 170 - 110$$<BR>\n" );
document.write( "$$f_1 + f_2 = 60 \quad \text{(Equation 1)}$$\r<BR>\n" );
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document.write( "### 2. Using the Median (Median = 35)\r<BR>\n" );
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document.write( "The median position is at $\frac{N}{2} = \frac{170}{2} = 85^{\text{th}}$ observation.\r<BR>\n" );
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document.write( "Since the median is $35$ (which falls in the $30-40$ class interval), the **median class** is **$30-40$**.\r<BR>\n" );
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document.write( "The formula for the median ($M$) of a grouped data set is:<BR>\n" );
document.write( "$$M = L + \left(\frac{\frac{N}{2} - C}{f_m}\right) \times w$$<BR>\n" );
document.write( "Where:<BR>\n" );
document.write( "* $L$ is the lower boundary of the median class ($30-40$), so $L = 30$.<BR>\n" );
document.write( "* $N$ is the total frequency, $N = 170$.<BR>\n" );
document.write( "* $C$ is the cumulative frequency of the class *before* the median class ($20-30$).<BR>\n" );
document.write( "    $$C = 10 + 20 + f_1 = 30 + f_1$$<BR>\n" );
document.write( "* $f_m$ is the frequency of the median class ($30-40$), so $f_m = 40$.<BR>\n" );
document.write( "* $w$ is the class width, $w = 10$.\r<BR>\n" );
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document.write( "Substitute the known values into the median formula:<BR>\n" );
document.write( "$$35 = 30 + \left(\frac{85 - (30 + f_1)}{40}\right) \times 10$$\r<BR>\n" );
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document.write( "Solve for $f_1$:<BR>\n" );
document.write( "$$35 - 30 = \frac{55 - f_1}{40} \times 10$$<BR>\n" );
document.write( "$$5 = \frac{55 - f_1}{4}$$<BR>\n" );
document.write( "$$5 \times 4 = 55 - f_1$$<BR>\n" );
document.write( "$$20 = 55 - f_1$$<BR>\n" );
document.write( "$$f_1 = 55 - 20$$<BR>\n" );
document.write( "$$\mathbf{f_1 = 35}$$\r<BR>\n" );
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document.write( "### 3. Finding $f_2$\r<BR>\n" );
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document.write( "Substitute $f_1 = 35$ into Equation 1:<BR>\n" );
document.write( "$$f_1 + f_2 = 60$$<BR>\n" );
document.write( "$$35 + f_2 = 60$$<BR>\n" );
document.write( "$$f_2 = 60 - 35$$<BR>\n" );
document.write( "$$\mathbf{f_2 = 25}$$\r<BR>\n" );
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document.write( "The missing frequencies are $\mathbf{f_1 = 35}$ and $\mathbf{f_2 = 25}$.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## (b) Calculating Mean, Interquartile Range, and Standard Deviation\r<BR>\n" );
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document.write( "Now we have the complete distribution: $N=170$.\r<BR>\n" );
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document.write( "| Score Class | Frequency ($f$) | Midpoint ($x$) | $f \cdot x$ | Cum. Freq. ($C_f$) |<BR>\n" );
document.write( "| :---: | :---: | :---: | :---: | :---: |<BR>\n" );
document.write( "| 0-10 | 10 | 5 | 50 | 10 |<BR>\n" );
document.write( "| 10-20 | 20 | 15 | 300 | 30 |<BR>\n" );
document.write( "| 20-30 | **35** ($f_1$) | 25 | 875 | 65 |<BR>\n" );
document.write( "| 30-40 | 40 | 35 | 1400 | 105 |<BR>\n" );
document.write( "| 40-50 | **25** ($f_2$) | 45 | 1125 | 130 |<BR>\n" );
document.write( "| 50-60 | 25 | 55 | 1375 | 155 |<BR>\n" );
document.write( "| 60-70 | 15 | 65 | 975 | 170 |<BR>\n" );
document.write( "| **Total** | **170** | | $\sum f x = 6100$ | |\r<BR>\n" );
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document.write( "### 1. Mean ($\bar{x}$)\r<BR>\n" );
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document.write( "The formula for the mean of grouped data is:<BR>\n" );
document.write( "$$\bar{x} = \frac{\sum f x}{N}$$<BR>\n" );
document.write( "$$\bar{x} = \frac{6100}{170}$$<BR>\n" );
document.write( "$$\bar{x} \approx \mathbf{35.88}$$\r<BR>\n" );
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document.write( "### 2. Interquartile Range (IQR)\r<BR>\n" );
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document.write( "$IQR = Q_3 - Q_1$\r<BR>\n" );
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document.write( "#### i. First Quartile ($Q_1$)<BR>\n" );
document.write( "$Q_1$ position is at $\frac{N}{4} = \frac{170}{4} = 42.5^{\text{th}}$ observation.<BR>\n" );
document.write( "The $42.5^{\text{th}}$ observation falls in the **$20-30$ class** ($C_f$ goes from 30 to 65).<BR>\n" );
document.write( "* $L = 20$<BR>\n" );
document.write( "* $C = 30$<BR>\n" );
document.write( "* $f_q = 35$<BR>\n" );
document.write( "* $w = 10$\r<BR>\n" );
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document.write( "$$Q_1 = L + \left(\frac{\frac{N}{4} - C}{f_q}\right) \times w = 20 + \left(\frac{42.5 - 30}{35}\right) \times 10$$<BR>\n" );
document.write( "$$Q_1 = 20 + \left(\frac{12.5}{35}\right) \times 10 \approx 20 + 3.5714$$<BR>\n" );
document.write( "$$\mathbf{Q_1 \approx 23.57}$$\r<BR>\n" );
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document.write( "#### ii. Third Quartile ($Q_3$)<BR>\n" );
document.write( "$Q_3$ position is at $\frac{3N}{4} = \frac{3 \times 170}{4} = 127.5^{\text{th}}$ observation.<BR>\n" );
document.write( "The $127.5^{\text{th}}$ observation falls in the **$40-50$ class** ($C_f$ goes from 105 to 130).<BR>\n" );
document.write( "* $L = 40$<BR>\n" );
document.write( "* $C = 105$<BR>\n" );
document.write( "* $f_q = 25$<BR>\n" );
document.write( "* $w = 10$\r<BR>\n" );
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document.write( "$$Q_3 = L + \left(\frac{\frac{3N}{4} - C}{f_q}\right) \times w = 40 + \left(\frac{127.5 - 105}{25}\right) \times 10$$<BR>\n" );
document.write( "$$Q_3 = 40 + \left(\frac{22.5}{25}\right) \times 10 = 40 + 0.9 \times 10 = 40 + 9$$<BR>\n" );
document.write( "$$\mathbf{Q_3 = 49}$$\r<BR>\n" );
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document.write( "#### iii. IQR<BR>\n" );
document.write( "$$IQR = Q_3 - Q_1 = 49 - 23.57 \approx \mathbf{25.43}$$\r<BR>\n" );
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document.write( "### 3. Standard Deviation ($\sigma$)\r<BR>\n" );
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document.write( "We need to calculate $\sum f x^2$:\r<BR>\n" );
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document.write( "| Class | $f$ | $x$ | $f \cdot x$ | $x^2$ | $f \cdot x^2$ |<BR>\n" );
document.write( "| :---: | :---: | :---: | :---: | :---: | :---: |<BR>\n" );
document.write( "| 0-10 | 10 | 5 | 50 | 25 | 250 |<BR>\n" );
document.write( "| 10-20 | 20 | 15 | 300 | 225 | 4500 |<BR>\n" );
document.write( "| 20-30 | 35 | 25 | 875 | 625 | 21875 |<BR>\n" );
document.write( "| 30-40 | 40 | 35 | 1400 | 1225 | 49000 |<BR>\n" );
document.write( "| 40-50 | 25 | 45 | 1125 | 2025 | 50625 |<BR>\n" );
document.write( "| 50-60 | 25 | 55 | 1375 | 3025 | 75625 |<BR>\n" );
document.write( "| 60-70 | 15 | 65 | 975 | 4225 | 63375 |<BR>\n" );
document.write( "| **Total** | **170** | | $\sum f x = 6100$ | | $\sum f x^2 = 265250$ |\r<BR>\n" );
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document.write( "The formula for the standard deviation ($\sigma$) is:<BR>\n" );
document.write( "$$\sigma = \sqrt{\frac{\sum f x^2}{N} - \bar{x}^2}$$<BR>\n" );
document.write( "$$\sigma = \sqrt{\frac{265250}{170} - (35.88235...)^2}$$<BR>\n" );
document.write( "$$\sigma = \sqrt{1560.2941 - 1287.545}$$<BR>\n" );
document.write( "$$\sigma = \sqrt{272.7491}$$<BR>\n" );
document.write( "$$\sigma \approx \mathbf{16.51}$$\r<BR>\n" );
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document.write( "### Summary of Results<BR>\n" );
document.write( "* **Mean ($\bar{x}$):** $35.88$<BR>\n" );
document.write( "* **Interquartile Range (IQR):** $25.43$<BR>\n" );
document.write( "* **Standard Deviation ($\sigma$):** $16.51$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## Explanation of Mean and Standard Deviation\r<BR>\n" );
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document.write( "### Meaning of the Mean ($\bar{x} \approx 35.88$)\r<BR>\n" );
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document.write( "The **mean** of $35.88$ represents the **average score** obtained by the students in Mr. Cheelo's Imionology class. This is the central tendency point—if the scores were redistributed equally among all 170 students, each student would have approximately $35.88$ points. This score falls within the middle range of the possible scores (0 to 70).\r<BR>\n" );
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document.write( "### Meaning of the Standard Deviation ($\sigma \approx 16.51$)\r<BR>\n" );
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document.write( "The **standard deviation** of $16.51$ is a measure of the **dispersion or variability** of the scores around the mean.<BR>\n" );
document.write( "* **Large Standard Deviation (like 16.51, which is $\approx 46\%$ of the mean):** Indicates that the scores are **widely spread out** from the average score. A significant number of students scored either much higher (e.g., above $35.88 + 16.51 = 52.39$) or much lower (e.g., below $35.88 - 16.51 = 19.37$).<BR>\n" );
document.write( "* **Conclusion for Mr. Cheelo's Class:** The high standard deviation suggests the class performance was **not homogeneous**. There was a large difference in performance between the highest and lowest-scoring students, indicating a wide range of academic ability or preparation in the class. </SPAN>\n" );
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