1. Observations from the first 25 days of the year showed that the following number of pedestrians used a particular crosswalk on each day:
19 132 25 78 62 101 84 32 99 71 43 56 64 81 59 67 43 97 105 91 31 66 76 87 125
(8 marks)
a. Construct a frequency distribution for this data using a lower limit for the first class of 15 and a class width of 20. Indicate the class limits, boundaries, midpoints, frequencies and cumulative frequencies.

Let's solve this step by step, constructing the required frequency distribution for the given pedestrian data. Here's the procedure:

### Data:
19, 132, 25, 78, 62, 101, 84, 32, 99, 71, 43, 56, 64, 81, 59, 67, 43, 97, 105, 91, 31, 66, 76, 87, 125

### Step 1: Define Class Limits

Given:
- **Lower limit of the first class** = 15
- **Class width** = 20

We will create class intervals with a width of 20 starting from 15.

| **Class Limits** | **Class Boundaries** | **Midpoint** | **Frequency** | **Cumulative Frequency** |
|------------------|----------------------|--------------|---------------|---------------------------|
| 15 - 34          | 14.5 - 34.5          | 24.5         |               |                           |
| 35 - 54          | 34.5 - 54.5          | 44.5         |               |                           |
| 55 - 74          | 54.5 - 74.5          | 64.5         |               |                           |
| 75 - 94          | 74.5 - 94.5          | 84.5         |               |                           |
| 95 - 114         | 94.5 - 114.5         | 104.5        |               |                           |
| 115 - 134        | 114.5 - 134.5        | 124.5        |               |                           |

### Step 2: Calculate the Midpoint

The midpoint is calculated as:

\[
\text{Midpoint} = \frac{\text{Lower Class Limit} + \text{Upper Class Limit}}{2}
\]

For example, for the first class (15 - 34):
\[
\text{Midpoint} = \frac{15 + 34}{2} = 24.5
\]

Similarly, we compute the midpoints for all classes.

### Step 3: Count the Frequency

Next, we tally how many data points fall into each class.

- For **15 - 34**: 19, 25, 32, 31 → Frequency = 4
- For **35 - 54**: 43, 43 → Frequency = 2
- For **55 - 74**: 62, 71, 64, 59, 67, 66 → Frequency = 6
- For **75 - 94**: 78, 84, 81, 91, 87 → Frequency = 5
- For **95 - 114**: 99, 97, 105, 101 → Frequency = 4
- For **115 - 134**: 132, 125 → Frequency = 2

### Step 4: Cumulative Frequency

The cumulative frequency is the running total of frequencies up to each class.

### Final Frequency Distribution Table:

| **Class Limits** | **Class Boundaries** | **Midpoint** | **Frequency** | **Cumulative Frequency** |
|------------------|----------------------|--------------|---------------|---------------------------|
| 15 - 34          | 14.5 - 34.5          | 24.5         | 4             | 4                         |
| 35 - 54          | 34.5 - 54.5          | 44.5         | 2             | 6                         |
| 55 - 74          | 54.5 - 74.5          | 64.5         | 6             | 12                        |
| 75 - 94          | 74.5 - 94.5          | 84.5         | 5             | 17                        |
| 95 - 114         | 94.5 - 114.5         | 104.5        | 4             | 21                        |
| 115 - 134        | 114.5 - 134.5        | 124.5        | 2             | 23                        |

### Explanation:
- **Class Limits**: These represent the intervals of the data.
- **Class Boundaries**: These are the actual boundaries used for continuous data.
- **Midpoint**: The central value of each class.
- **Frequency**: The count of data points in each class.
- **Cumulative Frequency**: The running total of frequencies as you progress through the classes.

Would you like more details or have any questions about this solution?

Here are 5 related questions you might explore:
1. What is the importance of class width in a frequency distribution?
2. How can we use this frequency distribution to plot a histogram?
3. What would change if the class width was increased to 25?
4. How does the cumulative frequency help in understanding data trends?
5. What is the difference between class boundaries and class limits?

**Tip**: When choosing class width, aim to balance between clarity (not too many classes) and precision (not too few classes).

Observations from the first 25 days of the year showed that the following number of pedestrians used a particular crosswalk on each day: 19 132 25 78 62 101 84 32 99 71 43 56 64 81 59 67 43 97 105 91 31 66 76 87 125 a. Construct a frequency distribution for this data using a lower limit for the first class of 15 and a class width of 20. Indicate the class limits, boundaries, midpoints, frequencies and cumulative frequencies.

Learn how to construct a frequency distribution for pedestrian data observed over 25 days using a class width of 20. This problem covers key statistical concepts, including class limits, boundaries, midpoints, and cumulative frequencies, with a step-by-step solution suitable for students in Grades 10-12.

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