Let's solve the problem step-by-step:
1. Given Information:
- Third Quartile (Q3): 25
- Quartile Deviation (QD): [tex]\( \frac{13}{2} \)[/tex]
2. Purpose:
- We need to find the coefficient of quartile deviation.
3. Relevant Formulas:
- Quartile Deviation (QD) is defined as:
[tex]\[
QD = \frac{Q3 - Q1}{2}
\][/tex]
- Coefficient of Quartile Deviation (CQD) is given by:
[tex]\[
CQD = \frac{Q3 - Q1}{Q3 + Q1}
\][/tex]
4. Steps to Solve:
- Firstly, the quartile deviation is given by:
[tex]\[
QD = \frac{13}{2} = 6.5
\][/tex]
- From the given quartile deviation, we can find the first quartile (Q1). Knowing that:
[tex]\[
QD = \frac{Q3 - Q1}{2}
\][/tex]
Hence:
[tex]\[
6.5 = \frac{25 - Q1}{2}
\][/tex]
- Multiply both sides by 2 to solve for [tex]\( Q1 \)[/tex]:
[tex]\[
13 = 25 - Q1
\][/tex]
- Isolate [tex]\( Q1 \)[/tex]:
[tex]\[
Q1 = 25 - 13
\][/tex]
[tex]\[
Q1 = 12
\][/tex]
- Now that we have both [tex]\( Q1 \)[/tex] and [tex]\( Q3 \)[/tex], we can calculate the coefficient of quartile deviation (CQD):
[tex]\[
CQD = \frac{Q3 - Q1}{Q3 + Q1}
\][/tex]
Substituting the values:
[tex]\[
CQD = \frac{25 - 12}{25 + 12}
\][/tex]
[tex]\[
CQD = \frac{13}{37}
\][/tex]
[tex]\[
CQD \approx 0.17567567567567569
\][/tex]
5. Summary:
- The first quartile Q1 is: 12.0
- The coefficient of quartile deviation is: 0.17567567567567569
This is the detailed solution to find the first quartile and the coefficient of quartile deviation.
