The table shows the annual compensation of 40 randomly chosen CEOs (millions of dollars).

5.35	1.93	24.93	8.19	14.57	5.42	3.26	4.12
192.00	16.07	23.62	7.60	10.32	15.58	2.49	5.56
82.81	17.55	27.08	6.38	8.88	5.98	1.73	5.94
26.67	13.00	19.41	4.63	9.86	4.53	0.79	2.23
31.47	12.83	20.16	6.16	8.89	4.46	1.38	4.05          Identify any unusual values.

Note: Round your answers to 2 decimal places.      The distribution is skewed to the right.

To identify unusual values in a dataset, we typically use the **IQR (Interquartile Range) method**. Here’s a step-by-step breakdown of the process to detect outliers:  

### 1. Sort the Data  
Sorted dataTo identify unusual values in a dataset, we typically use the **IQR (Interquartile Range) method**. Here’s a step-by-step breakdown of the process to detect outliers:  

### 1. Sort the Data  
Sorted   
0.79, 1.38, 1.73, 1.93, 2.23, 2.49, 3.26, 4.05, 4.12, 4.46, 4.53, 4.63, 5.35, 5.42, 5.56, 5.94, 5.98, 6.16, 6.38, 7.60, 8.19, 8.88, 8.89, 9.86, 10.32, 12.83, 13.00, 14.57, 15.58, 16.07, 17.55, 19.41, 20.16, 23.62, 24.93, 26.67, 27.08, 31.47, 82.81, 192.00  

### 2. Calculate the Quartiles  
- **Q1 (25th percentile):** Average of 10th and 11th values = (4.46 + 4.53) / 2 = **4.50**  
- **Q3 (75th percentile):** Average of 30th and 31st values = (16.07 + 17.55) / 2 = **16.81**

### 3. Calculate the IQR (Interquartile Range)  
\[
\text{IQR} = Q3 - Q1 = 16.81 - 4.50 = 12.31
\]

### 4. Find the Lower and Upper Fences  
Using the standard rule for outliers:  
- **Lower fence:** \( Q1 - 1.5 \times \text{IQR} = 4.50 - 1.5 \times 12.31 = 4.50 - 18.47 = -13.97 \)  
- **Upper fence:** \( Q3 + 1.5 \times \text{IQR} = 16.81 + 1.5 \times 12.31 = 16.81 + 18.47 = 35.28 \)  

### 5. Identify Unusual Values (Outliers)  
- Since the **lower fence** is negative, there are no low outliers.
- Any value **above 35.28** is considered an outlier.

From the data, the following values are above 35.28:
- **82.81**  
- **192.00**  

### 6. Conclusion  
The unusual (outlier) values in the dataset are:
- **82.81 million**  
- **192.00 million**

These values are much higher than the rest, consistent with the distribution being skewed to the right.

---

Do you have any questions or need further details?  
Here are five related questions to explore:  
1. How does the skewness affect the mean and median in this dataset?  
2. What is the impact of outliers on the standard deviation?  
3. How would the dataset change if we removed the outliers?  
4. Could a log transformation reduce skewness in this distribution?  
5. How do quartiles change if additional extreme outliers are introduced?

**Tip:** When working with skewed data, the **median** often provides a more representative measure of central tendency than the mean.

The table shows the annual compensation of 40 randomly chosen CEOs (millions of dollars).

5.35  1.93  24.93  8.19  14.57  5.42  3.26  4.12
192.00  16.07  23.62  7.60  10.32  15.58  2.49  5.56
82.81  17.55  27.08  6.38  8.88  5.98  1.73  5.94
26.67  13.00  19.41  4.63  9.86  4.53  0.79  2.23
31.47  12.83  20.16  6.16  8.89  4.46  1.38  4.05
Identify any unusual values. Note: Round your answers to 2 decimal places. The distribution is skewed to the right.

Analyze CEO compensation data to identify outliers using the IQR method. This problem demonstrates how to calculate quartiles, interquartile range, and detect unusual values based on upper and lower fences. Suitable for advanced high school students.

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