Calculate   the five-number summary   of the   given data.  Use the   approximation method.9,10,16,24,23,11,9,11,3,16,21,15,13,21,8Answer:

To find the five-number summary using the approximation method, we first need to sort the data, then identify the minimum, first quartile (Q1), median, third quartile (Q3), and maximum values.

### Step 1: Sort the data
Given dataTo find the five-number summary using the approximation method, we first need to sort the data, then identify the minimum, first quartile (Q1), median, third quartile (Q3), and maximum values.

### Step 1: Sort the data
Given   
\[ 9, 10, 16, 24, 23, 11, 9, 11, 3, 16, 21, 15, 13, 21, 8 \]

Sorted dataTo find the five-number summary using the approximation method, we first need to sort the data, then identify the minimum, first quartile (Q1), median, third quartile (Q3), and maximum values.

### Step 1: Sort the data
Given   
\[ 9, 10, 16, 24, 23, 11, 9, 11, 3, 16, 21, 15, 13, 21, 8 \]

Sorted   
\[ 3, 8, 9, 9, 10, 11, 11, 13, 15, 16, 16, 21, 21, 23, 24 \]

### Step 2: Calculate the five-number summary

1. **Minimum**: The smallest value in the data set.
   \[ \text{Minimum} = 3 \]

2. **First Quartile (Q1)**: The median of the lower half of the data (excluding the median if the number of observations is odd).
   \[
   Q1 = \text{Median of } \{3, 8, 9, 9, 10, 11, 11\} = 9
   \]

3. **Median**: The middle value of the data set.
   \[
   \text{Median} = 13
   \]

4. **Third Quartile (Q3)**: The median of the upper half of the data (excluding the median if the number of observations is odd).
   \[
   Q3 = \text{Median of } \{15, 16, 16, 21, 21, 23, 24\} = 21
   \]

5. **Maximum**: The largest value in the data set.
   \[ \text{Maximum} = 24 \]

### Final Five-Number Summary
\[
\text{Five-Number Summary} = (3, 9, 13, 21, 24)
\]

Would you like further details or have any questions?

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Here are some related questions to expand your understanding:

1. How would the five-number summary change if an additional value were added to the data set?
2. Can you explain why the first quartile (Q1) and third quartile (Q3) are important in understanding data spread?
3. How does the interquartile range (IQR) relate to the five-number summary?
4. What is the significance of outliers in the five-number summary?
5. How can the five-number summary be used in a box plot?

**Tip:** The interquartile range (IQR), calculated as \(Q3 - Q1\), helps in identifying outliers and understanding the spread of the middle 50% of the data.

Learn how to calculate the five-number summary using the approximation method for a given dataset (9, 10, 16, 24, 23, 11, 9, 11, 3, 16, 21, 15, 13, 21, 8). Understand the steps involved: sorting data, identifying minimum, first quartile (Q1), median, third quartile (Q3), and maximum values. Suitable for Grades 9-12.

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