Use the data set, which represents the fuel economies (mpg) of several motorcycles. Find the five number summary of the data set and draw a modified box-and-whisker plot for the data set. Use  asterisks (*) to represent the outliers. 
53 57 60 57 54 53 54 53 54 42 48 53 80 47 47 50 48 42 42 54 54 60


To find the five-number summary of the dataset and create a modified box-and-whisker plot, follow these steps:

### Data:
53, 57, 60, 57, 54, 53, 54, 53, 54, 42, 48, 53, 80, 47, 47, 50, 48, 42, 42, 54, 54, 60

### Step 1: Sort the Data
First, sort the data in ascending order:
42, 42, 42, 47, 47, 48, 48, 50, 53, 53, 53, 53, 54, 54, 54, 54, 57, 57, 60, 60, 80

### Step 2: Find the Five-Number Summary
The five-number summary consists of:
1. **Minimum (min)**: The smallest value in the dataset.
2. **First Quartile (Q1)**: The median of the lower half of the dataset.
3. **Median**: The middle value of the dataset.
4. **Third Quartile (Q3)**: The median of the upper half of the dataset.
5. **Maximum (max)**: The largest value in the dataset.

#### Minimum:
- The smallest value is **42**.

#### First Quartile (Q1):
- There are 11 values below the median:
  42, 42, 42, 47, 47, 48, 48, 50, 53, 53, 53
- The median of the lower half (between the 6th and 7th values) is **48**.

#### Median:
- The median is the middle value of the dataset. With 22 data points, the median is the average of the 11th and 12th values:
  \( \text{Median} = \frac{53 + 53}{2} = 53 \).

#### Third Quartile (Q3):
- There are 11 values above the median:
  53, 54, 54, 54, 54, 57, 57, 60, 60, 80
- The median of this half (between the 6th and 7th values) is **54**.

#### Maximum:
- The largest value is **80**.

### Step 3: Calculate Interquartile Range (IQR) and Identify Outliers
The interquartile range (IQR) is calculated as:
\[
IQR = Q3 - Q1 = 54 - 48 = 6
\]
To find potential outliers, we use the following rules:
- Lower Bound = \( Q1 - 1.5 \times IQR = 48 - 1.5 \times 6 = 48 - 9 = 39 \)
- Upper Bound = \( Q3 + 1.5 \times IQR = 54 + 1.5 \times 6 = 54 + 9 = 63 \)

Any values outside the range \( [39, 63] \) are considered outliers.

#### Outliers:
- Values less than 39: None
- Values greater than 63: **80** is an outlier.

### Step 4: Five-Number Summary
- **Minimum**: 42
- **First Quartile (Q1)**: 48
- **Median**: 53
- **Third Quartile (Q3)**: 54
- **Maximum** (without outliers): 60

### Step 5: Draw the Modified Box-and-Whisker Plot
To represent this with a modified box-and-whisker plot:
- The box will span from Q1 (48) to Q3 (54).
- The whiskers will extend from the minimum (42) to the maximum value within the range (60).
- The outlier (80) will be marked with an asterisk (*).

Here’s a text-based representation of the box-and-whisker plot:

```
|---*  80
|  |
|  |             |------|------|
|  |        42  48     53      54     60
```

Do you want further details or clarification?

---

### 5 Related Questions:
1. How do you calculate quartiles for a data set with an odd number of values?
2. What is the significance of the interquartile range (IQR)?
3. How do outliers affect the interpretation of a box-and-whisker plot?
4. Can the five-number summary be used for any type of data distribution?
5. What are the different methods for detecting outliers besides the IQR method?

**Tip**: Outliers can indicate interesting anomalies in your data set, but they can also skew results. It’s important to identify and consider their impact.

Use the data set, which represents the fuel economies (mpg) of several motorcycles. Find the five-number summary of the data set and draw a modified box-and-whisker plot for the data set. Use asterisks (*) to represent the outliers. 53 57 60 57 54 53 54 53 54 42 48 53 80 47 47 50 48 42 42 54 54 60

Discover how to calculate the five-number summary and create a modified box-and-whisker plot for a dataset representing motorcycle fuel economies. Learn to identify outliers using the interquartile range (IQR) method. Suitable for high school students (Grades 9-12).

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