Identify the lower class​ limits, upper class​ limits, class​ width, class​ midpoints, and class boundaries for the given frequency distribution. Also identify the number of individuals included in the summary.
Age (yr)                         Frequency
when award was won  
25 to 29	                          30
30 to 34	                          32
35 to 39	                          15
40 to 44	                           4
45 to 49	                           6
50 to 54	                           2
55 to 59	                           1

Let's break down the given frequency distribution of ages when an award was won. We will identify the **lower class limits**, **upper class limits**, **class width**, **class midpoints**, **class boundaries**, and the **total number of individuals** in the summary.

### 1. **Lower Class Limits**:
The lower class limit is the smallest value in each class.

- 25, 30, 35, 40, 45, 50, 55

### 2. **Upper Class Limits**:
The upper class limit is the largest value in each class.

- 29, 34, 39, 44, 49, 54, 59

### 3. **Class Width**:
Class width is the difference between consecutive lower class limits or upper class limits.
- \( 30 - 25 = 5 \)
- Class width = **5**

### 4. **Class Midpoints**:
The class midpoint is the average of the lower and upper class limits for each class:
\[
\text{Midpoint} = \frac{\text{Lower Class Limit} + \text{Upper Class Limit}}{2}
\]

- \( \frac{25 + 29}{2} = 27 \)
- \( \frac{30 + 34}{2} = 32 \)
- \( \frac{35 + 39}{2} = 37 \)
- \( \frac{40 + 44}{2} = 42 \)
- \( \frac{45 + 49}{2} = 47 \)
- \( \frac{50 + 54}{2} = 52 \)
- \( \frac{55 + 59}{2} = 57 \)

So, the class midpoints are:
- 27, 32, 37, 42, 47, 52, 57

### 5. **Class Boundaries**:
Class boundaries are values that separate classes without gaps between them. You find them by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit.

- For the class 25 to 29: Lower boundary = \( 25 - 0.5 = 24.5 \), Upper boundary = \( 29 + 0.5 = 29.5 \)
- For the class 30 to 34: Lower boundary = \( 30 - 0.5 = 29.5 \), Upper boundary = \( 34 + 0.5 = 34.5 \)
- For the class 35 to 39: Lower boundary = \( 35 - 0.5 = 34.5 \), Upper boundary = \( 39 + 0.5 = 39.5 \)
- For the class 40 to 44: Lower boundary = \( 40 - 0.5 = 39.5 \), Upper boundary = \( 44 + 0.5 = 44.5 \)
- For the class 45 to 49: Lower boundary = \( 45 - 0.5 = 44.5 \), Upper boundary = \( 49 + 0.5 = 49.5 \)
- For the class 50 to 54: Lower boundary = \( 50 - 0.5 = 49.5 \), Upper boundary = \( 54 + 0.5 = 54.5 \)
- For the class 55 to 59: Lower boundary = \( 55 - 0.5 = 54.5 \), Upper boundary = \( 59 + 0.5 = 59.5 \)

### 6. **Number of Individuals**:
To find the total number of individuals, sum the frequencies:

\[
30 + 32 + 15 + 4 + 6 + 2 + 1 = 90
\]

So, the total number of individuals is **90**.

### Summary:
- **Lower Class Limits**: 25, 30, 35, 40, 45, 50, 55
- **Upper Class Limits**: 29, 34, 39, 44, 49, 54, 59
- **Class Width**: 5
- **Class Midpoints**: 27, 32, 37, 42, 47, 52, 57
- **Class Boundaries**: 24.5–29.5, 29.5–34.5, 34.5–39.5, 39.5–44.5, 44.5–49.5, 49.5–54.5, 54.5–59.5
- **Number of Individuals**: 90

Do you want more details or have any questions?

### Related Questions:
1. How do you calculate the relative frequency from a frequency distribution?
2. What is the difference between class boundaries and class limits?
3. How do you construct a histogram using class boundaries?
4. How can we determine whether the distribution is symmetric or skewed from this data?
5. How would you calculate the cumulative frequency for each class?

**Tip**: Class width can always be checked by subtracting consecutive lower limits (or upper limits) to ensure uniform intervals.

Identify the lower class limits, upper class limits, class width, class midpoints, and class boundaries for the given frequency distribution. Also identify the number of individuals included in the summary.
Age (yr) Frequency when award was won
25 to 29 30
30 to 34 32
35 to 39 15
40 to 44 4
45 to 49 6
50 to 54 2
55 to 59 1

Learn how to identify lower and upper class limits, class width, midpoints, boundaries, and calculate the total number of individuals in a frequency distribution. This guide provides detailed steps and examples, making it perfect for students in Grades 10-12 studying statistics.

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