Certainly! Let's go through the steps to find the five-number summary for the given dataset:
Given data: 13, 20, 28, 17, 28, 30, 46, 18, 40, 28, 35, 39, 46, 45, 53
First, let's arrange the data in ascending order:
13, 17, 18, 20, 28, 28, 28, 30, 35, 39, 40, 45, 46, 46, 53
(a) To find the five-number summary, we need to identify the minimum, first quartile (Q1), median, third quartile (Q3), and maximum values.
1. Minimum: The smallest number in the dataset.
- Minimum = 13
2. First Quartile (Q1): This is the median of the lower half of the dataset (excluding the median if the dataset has an odd number of observations).
- Lower half: 13, 17, 18, 20, 28, 28, 28
- Q1 is the median of this lower half:
- Q1 = 24.0
3. Median: The middle value of the dataset.
- For the sorted dataset: 13, 17, 18, 20, 28, 28, 28, 30, 35, 39, 40, 45, 46, 46, 53
- Median = 30.0
4. Third Quartile (Q3): This is the median of the upper half of the dataset (excluding the median if the dataset has an odd number of observations).
- Upper half: 35, 39, 40, 45, 46, 46, 53
- Q3 is the median of this upper half:
- Q3 = 42.5
5. Maximum: The largest number in the dataset.
- Maximum = 53
Therefore, the five-number summary is:
- Minimum = 13
- First Quartile (Q1) = 24.0
- Median = 30.0
- Third Quartile (Q3) = 42.5
- Maximum = 53
So the five-number summary for this dataset is:
(13, 24.0, 30.0, 42.5, 53)
(b) To draw a box-and-whisker plot, follow these steps:
1. Draw a horizontal line and mark the scale based on the minimum and maximum values.
2. Draw vertical lines at the positions of the minimum (13), Q1 (24.0), median (30.0), Q3 (42.5), and maximum (53) values.
3. Draw a box from Q1 (24.0) to Q3 (42.5) and draw a vertical line inside the box at the median (30.0).
4. Draw “whiskers” from the minimum (13) to Q1 (24.0) and from Q3 (42.5) to the maximum (53).
Here's a simple representation:
```
|---|-----|-----------|--------|---|
13 24.0 30.0 42.5 53
```
In a box-and-whisker plot, the box represents the interquartile range (IQR), which contains the middle 50% of the data. The whiskers extend to the minimum and maximum values, providing visual insight into the spread and central tendency of the dataset.
