A sample of data consists of the values

12, 15, 17, 21, 21, 22, 23, 23, 23, 23, 24, 25, 28, 30

Using a modified box plot, determine which values, if any, are possible outliers. 

 Select all values that are possible outliers. 

Question 9 options:

12


15


17


25


28


30


there are no possible outliers

To determine if there are any possible outliers in the data using a modified box plot, we can follow these steps:

1. **Calculate the quartiles (Q1, Q2, Q3):**
   - **Q1 (first quartile)**: This is the median of the lower half of the data.
   - **Q2 (second quartile or median)**: This is the median of the entire dataset.
   - **Q3 (third quartile)**: This is the median of the upper half of the data.

2. **Find the Interquartile Range (IQR):**
   - \( IQR = Q3 - Q1 \)

3. **Determine the lower and upper bounds:**
   - **Lower bound**: \( Q1 - 1.5 \times IQR \)
   - **Upper bound**: \( Q3 + 1.5 \times IQR \)

4. **Identify outliers:**
   - Any data point below the lower bound or above the upper bound is considered a potential outlier.

### Step 1: Calculate Quartiles

Given dataTo determine if there are any possible outliers in the data using a modified box plot, we can follow these steps:

1. **Calculate the quartiles (Q1, Q2, Q3):**
   - **Q1 (first quartile)**: This is the median of the lower half of the data.
   - **Q2 (second quartile or median)**: This is the median of the entire dataset.
   - **Q3 (third quartile)**: This is the median of the upper half of the data.

2. **Find the Interquartile Range (IQR):**
   - \( IQR = Q3 - Q1 \)

3. **Determine the lower and upper bounds:**
   - **Lower bound**: \( Q1 - 1.5 \times IQR \)
   - **Upper bound**: \( Q3 + 1.5 \times IQR \)

4. **Identify outliers:**
   - Any data point below the lower bound or above the upper bound is considered a potential outlier.

### Step 1: Calculate Quartiles

Given  12, 15, 17, 21, 21, 22, 23, 23, 23, 23, 24, 25, 28, 30

- **Q2 (Median)**: 23 (since it is the average of the 7th and 8th values in the sorted list)
- **Q1**: Median of the first half (12, 15, 17, 21, 21, 22, 23) is 21.
- **Q3**: Median of the second half (23, 23, 23, 24, 25, 28, 30) is 24.

### Step 2: Calculate IQR

\( IQR = Q3 - Q1 = 24 - 21 = 3 \)

### Step 3: Determine the Lower and Upper Bounds

- **Lower bound**: \( Q1 - 1.5 \times IQR = 21 - 1.5 \times 3 = 21 - 4.5 = 16.5 \)
- **Upper bound**: \( Q3 + 1.5 \times IQR = 24 + 1.5 \times 3 = 24 + 4.5 = 28.5 \)

### Step 4: Identify Outliers

- Any value below 16.5 or above 28.5 is considered a possible outlier.

**Values in the data**:
- 12, 15, and 17 are below 16.5, so they are possible outliers.
- 30 is above 28.5, so it is a possible outlier.

**Conclusion**:
The possible outliers in the data set are **12, 15, 17, and 30**.

### Answer:
- 12
- 15
- 17
- 30

### Would you like further details or have any questions?

### Follow-up Questions:
1. What are the key differences between a modified box plot and a regular box plot?
2. How does the choice of the multiplier (1.5) affect the detection of outliers in box plots?
3. Can a value exactly at the boundary (like 16.5 or 28.5) be considered an outlier?
4. How would you interpret outliers in the context of a real-world data set?
5. What are some alternative methods to detect outliers in a data set?

### Tip:
When identifying outliers, it's helpful to plot the data using a box plot, as it visually represents the spread and identifies outliers easily.

Learn how to identify possible outliers in data using a modified box plot. Explore the steps to calculate quartiles, determine the Interquartile Range (IQR), and detect outliers. Example data set: 12, 15, 17, 21, 21, 22, 23, 23, 23, 23, 24, 25, 28, 30.

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