To find the lower quartile (Q1) of the given data set [tex]\( \{1, 3, 3, 4, 4, 4, 5, 6, 6, 8\} \)[/tex], follow these steps:
1. Organize the Data:
Ensure the data is in ascending order: [tex]\( \{1, 3, 3, 4, 4, 4, 5, 6, 6, 8\} \)[/tex].
2. Determine the Number of Data Points:
The data set contains 10 values.
3. Calculate the Position of the Lower Quartile:
The position of the lower quartile (Q1) is determined using the formula:
[tex]\[
Q1\_position = \frac{(n + 1)}{4}
\][/tex]
where [tex]\( n \)[/tex] is the number of data points. For this data set:
[tex]\[
Q1\_position = \frac{(10 + 1)}{4} = \frac{11}{4} = 2.75
\][/tex]
4. Determine the Value of Q1:
Since the position 2.75 is not an integer, we need to interpolate between the second and third data points in the ordered list.
- The 2nd data point is 3.
- The 3rd data point is 3.
To interpolate, we calculate:
[tex]\[
Q1 = \text{lower value} + (\text{fraction} \times \Delta \text{value})
\][/tex]
Here, the lower value is 3 (2nd data point), the fraction part is 0.75 (from 2.75), and the [tex]\(\Delta \text{value}\)[/tex] (difference between the 3rd and 2nd value) is 0.
So, the calculation will be:
[tex]\[
Q1 = 3 + 0.75 \times 0 = 3
\][/tex]
Therefore, the lower quartile (Q1) of the data set is:
[tex]\[
\boxed{3}
\][/tex]
