Question

Identify all values, if any, in the data set that would be considered outliers when creating a modified boxplot.
4 5 7 9 10 10 12 13 15 16 17 23 31
1)no outliers
2)31
3)4 and 31
4)4

Answer 1

To identify outliers using a modified boxplot (also known as a box-and-whisker plot), we typically use the interquartile range (IQR) method. Here’s how to find the outliers step-by-step:

1. **Order the data set**:
- Already ordered: 4, 5, 7, 9, 10, 10, 12, 13, 15, 16, 17, 23, 31

2. **Calculate the quartiles**:
- \( Q_1 \) (1st quartile): The median of the first half of the data.
- Data for \( Q_1 \): 4, 5, 7, 9, 10, 10
- Median of this set (middle two values): \( \frac{7 + 9}{2} = 8 \)
- So, \( Q_1 = 10 \).
- \( Q_3 \) (3rd quartile): The median of the second half of the data.
- Data for \( Q_3 \): 12, 13, 15, 16, 17, 23, 31
- Median of this set: \( \frac{16 + 17}{2} = 16.5 \)
- So, \( Q_3 = 17 \).

3. **Calculate the IQR**:
- \( \text{IQR} = Q_3 - Q_1 = 17 - 10 = 7 \).

4. **Determine the bounds for outliers**:
- Lower bound: \( Q_1 - 1.5 \times \text{IQR} = 10 - 1.5 \times 7 = 10 - 10.5 = -0.5 \)
- Upper bound: \( Q_3 + 1.5 \times \text{IQR} = 17 + 1.5 \times 7 = 17 + 10.5 = 27.5 \)

5. **Identify outliers**:
- Any data point below -0.5 or above 27.5 is considered an outlier.
- Analyzed data: 4, 5, 7, 9, 10, 10, 12, 13, 15, 16, 17, 23, 31
- The only value above the upper bound is **31**.

So the only outlier in this dataset is **31**.

Thus, the correct answer is: **2) 31**.