A cellular phone company monitors monthly phone usage. The following data represent the monthly phone use in minutes of one particular customer for the past 20 months. Use the given data to answer parts​ (a) and​ (b). 323

480

548

450

492

475

345

513

378

419

514

342

512

459

484

405

394

527

468

326
​(a) Determine the standard deviation and interquartile range of the data.


Let's solve for both the standard deviation and the interquartile range (IQR) of the given data set. Here’s the list of values representing monthly phone use in minutes:

**Data:**  
323, 480, 548, 450, 492, 475, 345, 513, 378, 419, 514, 342, 512, 459, 484, 405, 394, 527, 468, 326

### Step 1: Calculating the Standard Deviation

1. **Find the mean (average)**:
   \[
   \text{Mean} = \frac{\sum \text{Data}}{N} = \frac{323 + 480 + 548 + \ldots + 468 + 326}{20}
   \]

2. **Calculate each data point's deviation from the mean, square each deviation, and then find the average of those squared deviations**:
   \[
   \text{Variance} = \frac{\sum (\text{Data} - \text{Mean})^2}{N}
   \]

3. **Take the square root of the variance** to find the standard deviation.

### Step 2: Calculating the Interquartile Range (IQR)

1. **Sort the data** in ascending order.

2. **Find the first quartile (Q1)** and **third quartile (Q3)**. These are the 25th and 75th percentiles of the data, respectively.

3. **Subtract Q1 from Q3** to get the IQR:
   \[
   \text{IQR} = Q3 - Q1
   \]

Let me perform these calculations for you.The calculations yield the following results:

- **Standard Deviation:** \( 69.54 \) minutes (rounded to two decimal places)
- **Interquartile Range (IQR):** \( 107 \) minutes

These measures help in understanding the spread and variability of the monthly phone usage data.

Would you like more details or have any questions?

Here are five related questions to further your understanding:
1. How does the standard deviation help in assessing the consistency of monthly phone usage?
2. What can be inferred if the IQR is large in relation to the range of the data?
3. How would outliers in the data affect the calculation of standard deviation and IQR?
4. What is the significance of using population versus sample standard deviation?
5. How can quartiles help in understanding the distribution of data?

**Tip:** To reduce the impact of outliers on measures of variability, consider using the interquartile range instead of the standard deviation.

A cellular phone company monitors monthly phone usage. The following data represent the monthly phone use in minutes of one particular customer for the past 20 months. Use the given data to answer parts (a) and (b). Data: 323, 480, 548, 450, 492, 475, 345, 513, 378, 419, 514, 342, 512, 459, 484, 405, 394, 527, 468, 326 (a) Determine the standard deviation and interquartile range of the data.

Learn how to calculate the standard deviation and interquartile range (IQR) for monthly phone usage data over 20 months. This problem involves applying key statistical concepts, including the standard deviation and IQR formulas, making it suitable for Grades 9-12 students.

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