Here is a data set:
214	229	240	253
183	211	206	183
223	251	229	187
214	234	204	250
230	218	205	276
185	232	226	236
222	249	255	259


The goal is to construct a grouped frequency distribution table (GFDT) for this data set. The GFDT should have 10 classes with a "nice" class width. Each class should contain its lower class limit, and the lower class limits should all be multiples of the class width.

This problem is to determine what the class width and the first lower class limit should be.

What is the best class width for this data set?
optimal class width = 

What should be the first lower class limit?
1st lower class limit = 

To determine the optimal class width and the first lower class limit for a grouped frequency distribution table (GFDT), we'll follow these steps:

### Step 1: Find the Range of the Data
The range is calculated as:
\[
\text{Range} = \text{Maximum value} - \text{Minimum value}
\]
From the data set:

- Maximum value = 276
- Minimum value = 183

\[
\text{Range} = 276 - 183 = 93
\]

### Step 2: Determine the Class Width
To get a "nice" class width that makes the classes easy to interpret and ensures that each class contains its lower class limit as a multiple of the class width, we can follow this approach:

The class width \( w \) can be approximated by:
\[
w = \frac{\text{Range}}{\text{Number of classes}}
\]

Given that there should be 10 classes:
\[
w \approx \frac{93}{10} = 9.3
\]

We round this to a "nice" number, typically a value that is easy to work with (e.g., 5, 10, 15, etc.). The best "nice" number closest to 9.3 is **10**.

### Step 3: Determine the First Lower Class Limit
The first lower class limit should be a multiple of the class width. Since the smallest data value is 183, the first lower class limit should be the multiple of 10 that is less than or equal to 183. The closest multiple of 10 is **180**.

### Summary:
- **Optimal class width** = 10
- **First lower class limit** = 180

Would you like more details on constructing the entire grouped frequency distribution table? 

Here are five questions you might consider next:
1. How do you calculate the upper class limits for each class?
2. How would you determine the class boundaries in this grouped frequency distribution?
3. What steps are involved in tallying the frequencies for each class?
4. How can you represent this data visually using a histogram?
5. How does changing the class width affect the grouped frequency distribution?

**Tip:** When determining class widths, always ensure that the width is "nice" to make the data easier to interpret and use.

Learn how to construct a grouped frequency distribution table (GFDT) for a given data set. Determine the optimal class width and the first lower class limit with a step-by-step explanation. Suitable for high school students studying statistics.

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