Math Problem Statement

Find the five-number summary for the given data set: 3, 8, 8, 5, 2, 9, 8, 7, 9, 6, 9, 4, 1, 6, 2, 9, 8, 7, 7, 9. Draw a box-and-whisker plot and choose the correct plot (A or B).

Solution

Let's begin by finding the five-number summary for the given data set:

Data:

3, 8, 8, 5, 2, 9, 8, 7, 9, 6, 9, 4, 1, 6, 2, 9, 8, 7, 7, 9

First, we will sort the data in ascending order:

1, 2, 2, 3, 4, 5, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9

Now, we can calculate the five-number summary:

  1. Minimum (Min): The smallest value in the data set. Min=1\text{Min} = 1

  2. First Quartile (Q1): The median of the lower half of the data (excluding the median if the number of data points is odd). The lower half of the data is: 1,2,2,3,4,5,6,6,7,71, 2, 2, 3, 4, 5, 6, 6, 7, 7 The median of this half (Q1) is: Q1=5+62=5.5Q1 = \frac{5 + 6}{2} = 5.5

  3. Median (Q2): The middle value of the entire data set. The median of the full data set is: Median (Q2)=7\text{Median (Q2)} = 7

  4. Third Quartile (Q3): The median of the upper half of the data. The upper half of the data is: 7,8,8,8,8,9,9,9,9,97, 8, 8, 8, 8, 9, 9, 9, 9, 9 The median of this half (Q3) is: Q3=8+82=8Q3 = \frac{8 + 8}{2} = 8

  5. Maximum (Max): The largest value in the data set. Max=9\text{Max} = 9

The five-number summary is:

  • Min = 1
  • Q1 = 5.5
  • Median (Q2) = 7
  • Q3 = 8
  • Max = 9

Box-and-Whisker Plot:

Based on this summary, you can now choose the correct box-and-whisker plot. From the choices (A or B), I would need more context to determine which one is correct based on their labels. However, you can check which one shows:

  • The minimum at 1,
  • The lower quartile at 5.5,
  • The median at 7,
  • The upper quartile at 8,
  • And the maximum at 9.

Let me know if you want further details or if you have any questions.

Here are five related questions to expand on this topic:

  1. How do you calculate interquartile range (IQR) using the five-number summary?
  2. What does the box in the box-and-whisker plot represent?
  3. How do outliers affect the box-and-whisker plot?
  4. What are the advantages of using a box-and-whisker plot for data representation?
  5. How do you interpret the spread of data from a box-and-whisker plot?

Tip:

When calculating the quartiles, always ensure the data is sorted in ascending order before determining the median and quartiles.

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Math Problem Analysis

Mathematical Concepts

Descriptive Statistics
Quartiles
Box-and-Whisker Plot

Formulas

Five-number summary
Quartiles
Median

Theorems

The concept of quartiles and interquartile range

Suitable Grade Level

Grades 8-10