Certainly! Let's find the five-number summary and the Interquartile Range (IQR) for the given dataset [tex]\(3, 5, 6, 10, 7, 8, 9\)[/tex].
### Step-by-Step Solution
1. Sort the Data:
First, we need to sort the data in ascending order. The sorted data is:
[tex]\[
[3, 5, 6, 7, 8, 9, 10]
\][/tex]
2. Determine the Minimum and Maximum Values:
The minimum value is the smallest number in the sorted list, and the maximum value is the largest number in the sorted list.
[tex]\[
\text{Minimum} = 3
\][/tex]
[tex]\[
\text{Maximum} = 10
\][/tex]
3. Find the Median (Q2):
The median is the middle number in the sorted list. Since there are 7 numbers (an odd count), the median is the fourth number.
[tex]\[
\text{Median} = 7
\][/tex]
4. Divide the Data into Two Halves:
- The lower half consists of numbers before the median: [tex]\([3, 5, 6]\)[/tex]
- The upper half consists of numbers after the median: [tex]\([8, 9, 10]\)[/tex]
5. Find the First Quartile (Q1) and Third Quartile (Q3):
- Q1 is the median of the lower half of the data.
[tex]\[
\text{Lower half} = [3, 5, 6]
\][/tex]
Since there are 3 numbers, the median of this half is the second number.
[tex]\[
Q1 = 5.0
\][/tex]
- Q3 is the median of the upper half of the data.
[tex]\[
\text{Upper half} = [8, 9, 10]
\][/tex]
Since there are 3 numbers, the median of this half is the second number.
[tex]\[
Q3 = 9.0
\][/tex]
6. Calculate the Interquartile Range (IQR):
The IQR is the difference between the third quartile (Q3) and the first quartile (Q1).
[tex]\[
\text{IQR} = Q3 - Q1 = 9.0 - 5.0 = 4.0
\][/tex]
### Summary:
The five-number summary and IQR for the data [tex]\(3, 5, 6, 10, 7, 8, 9\)[/tex] are:
- Minimum: 3
- First Quartile (Q1): 5.0
- Median (Q2): 7.0
- Third Quartile (Q3): 9.0
- Maximum: 10
- Interquartile Range (IQR): 4.0
