To find the first (Q1) and third quartiles (Q3) of a set of numbers, you need to locate the values that divide the data set into four equal parts. Here is the step-by-step solution:
1. List the first 10 prime numbers in ascending order:
[tex]\[
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
\][/tex]
2. Determine the number of data points (n):
[tex]\[
n = 10
\][/tex]
3. Determine the positions of the first (Q1) and third quartiles (Q3):
- The position for Q1 (25th percentile) is given by:
[tex]\[
\text{Position of } Q1 = \frac{1}{4} (n + 1) = \frac{1}{4} (10 + 1) = \frac{1}{4} \times 11 = 2.75
\][/tex]
- The position for Q3 (75th percentile) is given by:
[tex]\[
\text{Position of } Q3 = \frac{3}{4} (n + 1) = \frac{3}{4} (10 + 1) = \frac{3}{4} \times 11 = 8.25
\][/tex]
4. Find the values at these positions using interpolation:
- For [tex]\( Q1 \)[/tex] (2.75th position):
- The 2nd position is the 2nd smallest value, which is [tex]\( 3 \)[/tex],
- The 3rd position is the 3rd smallest value, which is [tex]\( 5 \)[/tex].
- To find the value at the 2.75th position, interpolate between the 2nd and 3rd values:
[tex]\[
Q1 = 3 + 0.75 \times (5 - 3) = 3 + 0.75 \times 2 = 3 + 1.5 = 4.5
\][/tex]
- For [tex]\( Q3 \)[/tex] (8.25th position):
- The 8th position is the 8th smallest value, which is [tex]\( 19 \)[/tex],
- The 9th position is the 9th smallest value, which is [tex]\( 23 \)[/tex].
- To find the value at the 8.25th position, interpolate between the 8th and 9th values:
[tex]\[
Q3 = 19 + 0.25 \times (23 - 19) = 19 + 0.25 \times 4 = 19 + 1 = 20
\][/tex]
Therefore, the first quartile (Q1) is [tex]\( 4.5 \)[/tex] and the third quartile (Q3) is [tex]\( 20 \)[/tex].
First quartile (Q1) = 4.5
Third quartile (Q3) = 20
