Which choice below is a boxplot for the following distribution? 66, 62, 58, 52, 48, 46, 44, 34, 33, 31, 31, 30, 27, 25, 24, 21, 19 A. B. C. D. E. A. Boxplot A B. Boxplot B C. Boxplot C D. Boxplot D E. Boxplot E
The Correct Answer and Explanation is:
To determine which boxplot corresponds to the given data, we must first calculate the five-number summary:
Given Data (sorted):
19, 21, 24, 25, 27, 30, 31, 31, 33, 34, 44, 46, 48, 52, 58, 62, 66
There are 17 values.
Step 1: Find the Five-Number Summary
- Minimum: 19
- Maximum: 66
- Median (Q2): 9th value = 33
- Lower Quartile (Q1): Median of lower half (values 1–8): 19, 21, 24, 25, 27, 30, 31, 31
→ Median = average of 4th and 5th = (25 + 27)/2 = 26 - Upper Quartile (Q3): Median of upper half (values 10–17): 34, 44, 46, 48, 52, 58, 62, 66
→ Median = average of 4th and 5th = (48 + 52)/2 = 50
Five-Number Summary:
- Min = 19
- Q1 = 26
- Median = 33
- Q3 = 50
- Max = 66
Step 2: Compare with Boxplots
We now match this summary with the boxplots in the image.
- Look for a plot where:
- The box starts around 26 (Q1)
- The line in the box is at 33 (median)
- The box ends around 50 (Q3)
- The whiskers go from 19 to 66
Upon inspection, Boxplot D matches this five-number summary accurately.
✅ Correct Answer: D. Boxplot D
Explanation
To find the correct boxplot for a data set, we need to summarize the distribution using a five-number summary: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum. For the given distribution, we first sort the data in ascending order. With 17 numbers, the middle value (the 9th) is the median. The lower and upper halves (excluding the median if odd) give us the quartiles.
For this data, the minimum is 19 and the maximum is 66. The median (Q2) is the 9th value, which is 33. Q1, the median of the lower half (first 8 values), is (25 + 27)/2 = 26. Q3, the median of the upper half (last 8 values), is (48 + 52)/2 = 50.
We then compare these calculated summary points to each boxplot option. The correct boxplot must have a box stretching from 26 to 50 with a line at 33, and whiskers extending to 19 and 66. Only Boxplot D fits all these criteria.
This method provides a visual summary of data distribution, helping identify the center, spread, and any skewness.
