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To tackle this problem, we need to construct a frequency distribution and a histogram for the given data using 6 classes. Here's the process:

1. **Determine the Range of the Data:**
   The given data is: \(6, 9, 7, 4, 5, 8, 4, 11, 10, 8, 6, 5, 7, 6, 8, 7, 6, 3, 11\)

   - Minimum value: \(3\)
   - Maximum value: \(11\)
   - Range = \(11 - 3 = 8\)

2. **Determine the Class Width:**
   To find the class width, divide the range by the number of classes and round up to the nearest whole number:
   - Class width = \(\lceil \frac{8}{6} \rceil = 2\)

3. **Determine the Class Limits:**
   The classes can be determined starting from the minimum value:
   - Class 1: \(3 - 4\)
   - Class 2: \(5 - 6\)
   - Class 3: \(7 - 8\)
   - Class 4: \(9 - 10\)
   - Class 5: \(11 - 12\)

4. **Construct the Frequency Distribution:**
   Tally the frequencies for each class.

   | Class  | Frequency | Midpoint |
   |--------|-----------|----------|
   | 3 - 4  | 3         | 3.5      |
   | 5 - 6  | 6         | 5.5      |
   | 7 - 8  | 6         | 7.5      |
   | 9 - 10 | 2         | 9.5      |
   | 11 - 12| 2         | 11.5     |

5. **Describe the Shape of the Histogram:**
   - Given the frequencies, the histogram shape can be described. Observing the dataTo tackle this problem, we need to construct a frequency distribution and a histogram for the given data using 6 classes. Here's the process:

1. **Determine the Range of the Data:**
   The given data is: \(6, 9, 7, 4, 5, 8, 4, 11, 10, 8, 6, 5, 7, 6, 8, 7, 6, 3, 11\)

   - Minimum value: \(3\)
   - Maximum value: \(11\)
   - Range = \(11 - 3 = 8\)

2. **Determine the Class Width:**
   To find the class width, divide the range by the number of classes and round up to the nearest whole number:
   - Class width = \(\lceil \frac{8}{6} \rceil = 2\)

3. **Determine the Class Limits:**
   The classes can be determined starting from the minimum value:
   - Class 1: \(3 - 4\)
   - Class 2: \(5 - 6\)
   - Class 3: \(7 - 8\)
   - Class 4: \(9 - 10\)
   - Class 5: \(11 - 12\)

4. **Construct the Frequency Distribution:**
   Tally the frequencies for each class.

   | Class  | Frequency | Midpoint |
   |--------|-----------|----------|
   | 3 - 4  | 3         | 3.5      |
   | 5 - 6  | 6         | 5.5      |
   | 7 - 8  | 6         | 7.5      |
   | 9 - 10 | 2         | 9.5      |
   | 11 - 12| 2         | 11.5     |

5. **Describe the Shape of the Histogram:**
   - Given the frequencies, the histogram shape can be described. Observing the 
     - The distribution appears to be approximately symmetric but slightly positively skewed, since the majority of the data is clustered towards the lower end with a few higher values.

Here is the frequency distribution and histogram details:

### Frequency Distribution:
| Class  | Frequency | Midpoint |
|--------|-----------|----------|
| 3 - 4  | 3         | 3.5      |
| 5 - 6  | 6         | 5.5      |
| 7 - 8  | 6         | 7.5      |
| 9 - 10 | 2         | 9.5      |
| 11 - 12| 2         | 11.5     |

### Histogram Shape:
- The histogram is slightly positively skewed.

Would you like further details or have any questions?

### Relative Questions:
1. How is the class width calculated and why is it important?
2. What is the significance of using midpoints in frequency distributions?
3. Can you explain the process of creating a histogram from a frequency distribution?
4. How do different shapes of histograms (symmetric, skewed, uniform) help in understanding data distribution?
5. What are some potential sources of error when constructing a frequency distribution and histogram?

### Tip:
When constructing histograms, always ensure that the class widths are consistent to maintain accuracy in data representation.

The numbers of days 20 patients remained hospitalized are shown below: 6, 9, 7, 4, 5, 8, 4, 11, 10, 8, 6, 5, 7, 6, 8, 7, 6, 3, 11. Construct a frequency distribution and a frequency histogram of the data using 6 classes. Describe the shape of the histogram as symmetric, uniform, negatively skewed, positively skewed, or none of these.

Learn how to construct a frequency distribution and histogram for the number of days 20 patients remained hospitalized. The dataset uses 6 classes, and the shape of the histogram is analyzed to determine if it's symmetric, skewed, or uniform. Suitable for Grades 9-12 students learning data analysis.

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