Answer :
Let's solve the problem step by step to determine the median, standard deviation, and coefficient of variation for the given weight data.
### Step 1: Determine the Midpoints of Each Weight Range
First, we need to find the midpoint of each weight range. These midpoints represent the average value of each weight range:
- For [tex]\(10 < x < 20\)[/tex]: midpoint is [tex]\(\frac{10 + 20}{2} = 15\)[/tex]
- For [tex]\(20 < x < 30\)[/tex]: midpoint is [tex]\(\frac{20 + 30}{2} = 25\)[/tex]
- For [tex]\(30 < x < 40\)[/tex]: midpoint is [tex]\(\frac{30 + 40}{2} = 35\)[/tex]
- For [tex]\(40 < x < 50\)[/tex]: midpoint is [tex]\(\frac{40 + 50}{2} = 45\)[/tex]
So, the midpoints are: [tex]\(15, 25, 35, 45\)[/tex].
### Step 2: Organize the Data
We then organize the data into two lists:
- Midpoints [tex]\([15, 25, 35, 45]\)[/tex]
- Frequencies [tex]\([10, 8, 5, 7]\)[/tex]
### Step 3: Calculate the Median
To calculate the median, we need to find the cumulative frequency and locate the median class.
- The cumulative frequencies are calculated as follows:
- [tex]\(10\)[/tex] (first class)
- [tex]\(10 + 8 = 18\)[/tex] (second class)
- [tex]\(18 + 5 = 23\)[/tex] (third class)
- [tex]\(23 + 7 = 30\)[/tex] (fourth class)
- The total number of bags [tex]\(n = 10 + 8 + 5 + 7 = 30\)[/tex].
- The median class is the [tex]\(\frac{n}{2}\)[/tex]-th or [tex]\(\frac{30}{2} = 15\)[/tex]-th item.
We find the median class using the cumulative frequency:
- The 15th item falls in the second class since the cumulative frequency of the first class is 10, and the cumulative frequency of the second class is 18.
Therefore, the median weight is the midpoint of the second class, which is [tex]\(25\)[/tex].
### Step 4: Calculate the Mean
The mean weight is calculated using the weighted average formula:
[tex]\[ \text{Mean} = \frac{\sum (x_i \cdot f_i)}{\sum f_i} \][/tex]
Using the given midpoints and frequencies:
[tex]\[ \text{Mean} = \frac{(15 \cdot 10) + (25 \cdot 8) + (35 \cdot 5) + (45 \cdot 7)}{30} = \frac{150 + 200 + 175 + 315}{30} = \frac{840}{30} = 28 \][/tex]
### Step 5: Calculate the Standard Deviation
The standard deviation [tex]\(\sigma\)[/tex] is calculated using the formula:
[tex]\[ \text{Variance} = \frac{\sum f_i (x_i - \text{mean})^2}{\sum f_i} \][/tex]
[tex]\[ \text{Standard Deviation} = \sqrt{\text{Variance}} \][/tex]
Using our midpoints, frequencies, and mean:
- [tex]\( \sum f_i = 30 \)[/tex]
- Mean = 28
[tex]\[ \text{Variance} = \frac{10 (15 - 28)^2 + 8 (25 - 28)^2 + 5 (35 - 28)^2 + 7 (45 - 28)^2}{30} = \frac{10 \cdot 169 + 8 \cdot 9 + 5 \cdot 49 + 7 \cdot 289}{30} = \frac{1690 + 72 + 245 + 2023}{30} = \frac{4030}{30} \approx 134.33 \][/tex]
So, the standard deviation:
[tex]\[ \text{Standard Deviation} = \sqrt{134.33} \approx 11.59 \][/tex]
### Step 6: Calculate the Coefficient of Variation
The coefficient of variation (CV) is calculated as follows:
[tex]\[ \text{CV} = \left(\frac{\text{Standard Deviation}}{\text{Mean}}\right) \times 100 = \left(\frac{11.59}{28}\right) \times 100 \approx 41.39\% \][/tex]
### Conclusion
Finally, the results are:
i) Median: [tex]\(25\)[/tex]
ii) Standard Deviation: [tex]\(11.59\)[/tex] (approximately)
iii) Coefficient of Variation: [tex]\(41.39\%\)[/tex] (approximately)
### Step 1: Determine the Midpoints of Each Weight Range
First, we need to find the midpoint of each weight range. These midpoints represent the average value of each weight range:
- For [tex]\(10 < x < 20\)[/tex]: midpoint is [tex]\(\frac{10 + 20}{2} = 15\)[/tex]
- For [tex]\(20 < x < 30\)[/tex]: midpoint is [tex]\(\frac{20 + 30}{2} = 25\)[/tex]
- For [tex]\(30 < x < 40\)[/tex]: midpoint is [tex]\(\frac{30 + 40}{2} = 35\)[/tex]
- For [tex]\(40 < x < 50\)[/tex]: midpoint is [tex]\(\frac{40 + 50}{2} = 45\)[/tex]
So, the midpoints are: [tex]\(15, 25, 35, 45\)[/tex].
### Step 2: Organize the Data
We then organize the data into two lists:
- Midpoints [tex]\([15, 25, 35, 45]\)[/tex]
- Frequencies [tex]\([10, 8, 5, 7]\)[/tex]
### Step 3: Calculate the Median
To calculate the median, we need to find the cumulative frequency and locate the median class.
- The cumulative frequencies are calculated as follows:
- [tex]\(10\)[/tex] (first class)
- [tex]\(10 + 8 = 18\)[/tex] (second class)
- [tex]\(18 + 5 = 23\)[/tex] (third class)
- [tex]\(23 + 7 = 30\)[/tex] (fourth class)
- The total number of bags [tex]\(n = 10 + 8 + 5 + 7 = 30\)[/tex].
- The median class is the [tex]\(\frac{n}{2}\)[/tex]-th or [tex]\(\frac{30}{2} = 15\)[/tex]-th item.
We find the median class using the cumulative frequency:
- The 15th item falls in the second class since the cumulative frequency of the first class is 10, and the cumulative frequency of the second class is 18.
Therefore, the median weight is the midpoint of the second class, which is [tex]\(25\)[/tex].
### Step 4: Calculate the Mean
The mean weight is calculated using the weighted average formula:
[tex]\[ \text{Mean} = \frac{\sum (x_i \cdot f_i)}{\sum f_i} \][/tex]
Using the given midpoints and frequencies:
[tex]\[ \text{Mean} = \frac{(15 \cdot 10) + (25 \cdot 8) + (35 \cdot 5) + (45 \cdot 7)}{30} = \frac{150 + 200 + 175 + 315}{30} = \frac{840}{30} = 28 \][/tex]
### Step 5: Calculate the Standard Deviation
The standard deviation [tex]\(\sigma\)[/tex] is calculated using the formula:
[tex]\[ \text{Variance} = \frac{\sum f_i (x_i - \text{mean})^2}{\sum f_i} \][/tex]
[tex]\[ \text{Standard Deviation} = \sqrt{\text{Variance}} \][/tex]
Using our midpoints, frequencies, and mean:
- [tex]\( \sum f_i = 30 \)[/tex]
- Mean = 28
[tex]\[ \text{Variance} = \frac{10 (15 - 28)^2 + 8 (25 - 28)^2 + 5 (35 - 28)^2 + 7 (45 - 28)^2}{30} = \frac{10 \cdot 169 + 8 \cdot 9 + 5 \cdot 49 + 7 \cdot 289}{30} = \frac{1690 + 72 + 245 + 2023}{30} = \frac{4030}{30} \approx 134.33 \][/tex]
So, the standard deviation:
[tex]\[ \text{Standard Deviation} = \sqrt{134.33} \approx 11.59 \][/tex]
### Step 6: Calculate the Coefficient of Variation
The coefficient of variation (CV) is calculated as follows:
[tex]\[ \text{CV} = \left(\frac{\text{Standard Deviation}}{\text{Mean}}\right) \times 100 = \left(\frac{11.59}{28}\right) \times 100 \approx 41.39\% \][/tex]
### Conclusion
Finally, the results are:
i) Median: [tex]\(25\)[/tex]
ii) Standard Deviation: [tex]\(11.59\)[/tex] (approximately)
iii) Coefficient of Variation: [tex]\(41.39\%\)[/tex] (approximately)
