What is the interquartile range for the data set
8,1,7,3,7,2,6,7,9
A. 3.5
B 5
C 5.6
D 7
The Correct Answer and Explanation is :
To find the interquartile range (IQR) of the given data set, we first need to organize the data in ascending order. The data set provided is:
[ 8, 1, 7, 3, 7, 2, 6, 7, 9 ]
Step 1: Arrange the Data
Arranging the numbers in ascending order:
[ 1, 2, 3, 6, 7, 7, 7, 8, 9 ]
Step 2: Find the Quartiles
Next, we need to find the first quartile (Q1), the second quartile (Q2, or median), and the third quartile (Q3).
- Median (Q2): The median is the middle number in a sorted data set. With nine numbers, the median is the fifth number:
[
Q2 = 7
] - First Quartile (Q1): Q1 is the median of the first half of the data (excluding the median). The first half of the data is:
[ 1, 2, 3, 6 ]
The median of this subset (the average of the second and third numbers) is:
[
Q1 = \frac{2 + 3}{2} = 2.5
] - Third Quartile (Q3): Q3 is the median of the second half of the data (also excluding the median). The second half of the data is:
[ 7, 7, 8, 9 ]
The median of this subset (the average of the second and third numbers) is:
[
Q3 = \frac{7 + 8}{2} = 7.5
]
Step 3: Calculate the Interquartile Range
Now that we have Q1 and Q3, we can calculate the IQR:
[
\text{IQR} = Q3 – Q1 = 7.5 – 2.5 = 5
]
Conclusion
The interquartile range for the data set is 5. Therefore, the correct answer is B. 5.
The IQR is a measure of statistical dispersion and is particularly useful because it is not influenced by outliers. It gives us an idea of the range within which the central 50% of the data lies. In this case, with an IQR of 5, we can conclude that the middle 50% of the data points are concentrated within a span of 5 units.