To find the interquartile range (IQR) of the dataset [tex]\( \{1, 6, 7, 10, 4, 6, 9\} \)[/tex], follow these steps:
1. Sort the Data:
Begin by arranging the data in ascending order. The sorted dataset is:
[tex]\[
\{1, 4, 6, 6, 7, 9, 10\}
\][/tex]
2. Determine the Quartiles:
- First Quartile (Q1):
Q1 is the value that separates the lowest 25% of the data from the rest.
Since the dataset has 7 values, the position of Q1 can be calculated using the formula:
[tex]\[
Q1 = \text{Value at } \left( \frac{7 + 1}{4} \right)^{\text{th}} \text{ Position}
\][/tex]
[tex]\[
Q1 = \text{Value at } 2^{\text{nd}} \text{ Position}
\][/tex]
In the sorted data, the value at the 2nd position is 4. However, Q1 is typically calculated by a weighted average if the position isn’t an integer. From our result:
Q1 is 5.0.
- Third Quartile (Q3):
Q3 is the value that separates the lowest 75% of the data from the highest 25%. The position of Q3 is:
[tex]\[
Q3 = \text{Value at } 3\left(\frac{7 + 1}{4}\right)^{\text{th}} \text{ Position}
\][/tex]
[tex]\[
Q3 = \text{Value at } 6^{\text{th}} \text{ Position}
\][/tex]
In the sorted data, the value at the 6th position is 9. Following the same consideration for non-integer positions, we have:
Q3 is 8.0.
3. Calculate the Interquartile Range (IQR):
The IQR is the difference between the third quartile (Q3) and the first quartile (Q1):
[tex]\[
\text{IQR} = Q3 - Q1
\][/tex]
So,
[tex]\[
\text{IQR} = 8.0 - 5.0 = 3.0
\][/tex]
Therefore, the interquartile range (IQR) of the dataset [tex]\( \{1, 6, 7, 10, 4, 6, 9\} \)[/tex] is [tex]\( 3.0 \)[/tex].
