Construct a frequency distribution for the given data set using 6 classes. In the table, include the midpoints, relative frequencies, and cumulative frequencies. Which class has the greatest frequency and which has the least frequency?

188

419

147

400

290

297

196

30

394

532

155

126

468

179

249

127

206 113 135

72 122 380 260 114

405

317

289

206

394

Complete the table, starting with the lowest class limit. Use the minimum data entry as the lower limit of the first class (Type integers or decimals. Round the class limits to the nearest whole number. Round all other values to the nearest thousandth as needed.)

To construct a frequency distribution table with 6 classes for the given data set, we'll follow these steps:

1. **Determine the range**: Find the difference between the maximum and minimum values.
2. **Determine the class width**: Divide the range by the number of classes and round up.
3. **Create the classes**: Start with the minimum value and add the class width to determine the upper limit of each class.
4. **Tally the data into classes**: Count how many data points fall into each class.
5. **Calculate midpoints**: For each class, find the midpoint by averaging the lower and upper limits.
6. **Calculate relative frequencies**: Divide the frequency of each class by the total number of data points.
7. **Calculate cumulative frequencies**: Add the frequencies of the current class and all previous classes.

### Step 1: Determine the Range

- **Minimum value** = 30
- **Maximum value** = 532

\[
\text{Range} = 532 - 30 = 502
\]

### Step 2: Determine the Class Width

\[
\text{Class width} = \frac{\text{Range}}{\text{Number of classes}} = \frac{502}{6} \approx 83.67 \Rightarrow \text{Round up to 84}
\]

### Step 3: Create the Classes

Starting with the minimum value (30), we'll create 6 classes with a width of 84.

| **Class** | **Lower Limit** | **Upper Limit** |
|-----------|-----------------|-----------------|
| 1         | 30              | 113             |
| 2         | 114             | 197             |
| 3         | 198             | 281             |
| 4         | 282             | 365             |
| 5         | 366             | 449             |
| 6         | 450             | 533             |

### Step 4: Tally the Data

Count how many data points fall into each class.

### Step 5: Calculate Midpoints

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

### Step 6: Calculate Relative Frequencies

\[
\text{Relative Frequency} = \frac{\text{Class Frequency}}{\text{Total Data Points}}
\]

### Step 7: Calculate Cumulative Frequencies

Now let's build the table:

| **Class** | **Frequency** | **Midpoint** | **Relative Frequency** | **Cumulative Frequency** |
|-----------|---------------|--------------|------------------------|--------------------------|
| 30 - 113  | 4             | 71.5         | 0.133                  | 4                        |
| 114 - 197 | 10            | 155.5        | 0.333                  | 14                       |
| 198 - 281 | 6             | 239.5        | 0.200                  | 20                       |
| 282 - 365 | 6             | 323.5        | 0.200                  | 26                       |
| 366 - 449 | 7             | 407.5        | 0.233                  | 33                       |
| 450 - 533 | 2             | 491.5        | 0.067                  | 35                       |

- **Total Data Points** = 35

### Final Results:

- **Class with the Greatest Frequency**: 114 - 197 (Frequency = 10)
- **Class with the Least Frequency**: 450 - 533 (Frequency = 2)

Would you like more details or have any questions? Here are some related questions you might find helpful:

1. How do you interpret the relative frequency of a class?
2. What is the significance of cumulative frequency in data analysis?
3. Can you describe how the midpoint of a class is used in data analysis?
4. How does increasing the number of classes affect the frequency distribution?
5. What might cause the frequency of a class to be particularly high or low?

**Tip:** Always ensure the total of relative frequencies equals 1 (or 100%) to check your calculations.

Learn how to construct a frequency distribution table with 6 classes for a given dataset. Includes steps to determine range, class width, create classes, calculate midpoints, relative frequencies, and cumulative frequencies. Ideal for advanced high school students studying statistics.

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