Following are the number of items of similar type produced in a factory during the last 50 days. 21 22 17 23 27 15 16 22 15 23 24 25 36 19 14 21 24 25 14 18 20 31 22 19 18 20 21 20 36 18 21 20 31 22 19 18 20 20 24 35 25 26 19 32 22 26 25 26 27 22 Arrange these observations into a frequency distribution with both inclusive and exclusive class intervals choosing a suitable number of classes.

To arrange the given observations into a frequency distribution, both with inclusive and exclusive class intervals, we need to follow these steps:

1. **Find the Range**:  
   - The range is the difference between the maximum and minimum values.
   - Maximum value = 36
   - Minimum value = 14
   - Range = 36 - 14 = 22

2. **Determine the Number of Classes**:  
   - The number of classes is usually determined by the Sturges' formula:
     \[
     \text{Number of classes} = 1 + 3.322 \times \log_{10}(n)
     \]
     where \( n \) is the number of observations.
   - For \( n = 50 \):
     \[
     \text{Number of classes} = 1 + 3.322 \times \log_{10}(50) \approx 1 + 3.322 \times 1.699 \approx 6.64 \approx 7
     \]
   - So, we will use 7 classes.

3. **Determine the Class Width**:  
   - Class width = \(\frac{\text{Range}}{\text{Number of classes}} = \frac{22}{7} \approx 3.14 \)
   - Rounding up to the nearest whole number, we get a class width of 4.

4. **Form the Class Intervals**:
   - For **inclusive** class intervals, each class will be of the form \([a, b]\).
   - For **exclusive** class intervals, each class will be of the form \([a, b)\).

### Inclusive Class Intervals:
Let's start from the minimum value (14) and create intervals of width 4.

- 14 - 17
- 18 - 21
- 22 - 25
- 26 - 29
- 30 - 33
- 34 - 37

Now, we tally the observations into these intervals:

| Class Interval | Frequency |
|----------------|-----------|
| 14 - 17        | 7         |
| 18 - 21        | 12        |
| 22 - 25        | 16        |
| 26 - 29        | 7         |
| 30 - 33        | 4         |
| 34 - 37        | 4         |

### Exclusive Class Intervals:
For exclusive class intervals, the first interval will be [14, 18) and so on:

- 14 - 18
- 18 - 22
- 22 - 26
- 26 - 30
- 30 - 34
- 34 - 38

Now, tally the observations:

| Class Interval | Frequency |
|----------------|-----------|
| 14 - 18        | 12        |
| 18 - 22        | 19        |
| 22 - 26        | 16        |
| 26 - 30        | 7         |
| 30 - 34        | 4         |
| 34 - 38        | 2         |

### Summary

- **Inclusive Class Intervals**:
  - 14 - 17: 7
  - 18 - 21: 12
  - 22 - 25: 16
  - 26 - 29: 7
  - 30 - 33: 4
  - 34 - 37: 4

- **Exclusive Class Intervals**:
  - 14 - 18: 12
  - 18 - 22: 19
  - 22 - 26: 16
  - 26 - 30: 7
  - 30 - 34: 4
  - 34 - 38: 2

Would you like more details or have any questions?

Here are some related questions to consider:

1. What are the advantages of using inclusive class intervals over exclusive ones?
2. How would the frequency distribution change if the class width were different?
3. How do you calculate the cumulative frequency for the intervals?
4. What is the importance of choosing an appropriate number of classes?
5. How would you graphically represent this frequency distribution?

**Tip**: When constructing frequency distributions, always consider the nature of your data and choose class intervals that effectively summarize the data without losing important details.

Explore how to analyze factory production data using frequency distribution and class intervals. Learn to organize observations into inclusive and exclusive class intervals, using Sturges' formula to determine class width. Suitable for high school students studying statistics.

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