To find the First Quartile (Q1) in a symmetric distribution given that the Interquartile Range (IQR) is 10 and the median is 10, we can follow these steps:
1. Understand the Interquartile Range (IQR): The IQR is the range within which the middle 50% of the data lies. It is the difference between the Third Quartile (Q3) and the First Quartile (Q1):
[tex]\[
IQR = Q3 - Q1
\][/tex]
Given:
[tex]\[
IQR = 10
\][/tex]
2. Utilize the Median in a Symmetric Distribution: In a symmetric distribution, the median is the central point. This means that the distances from the median to Q1 and from the median to Q3 are equal. Therefore:
[tex]\[
Q3 - \text{median} = \text{median} - Q1
\][/tex]
Given that the median is also 10, we can substitute the median into our equation.
3. Express Q3 in Terms of Q1: Since the distances are equal:
[tex]\[
Q3 = \text{median} + (\text{median} - Q1)
\][/tex]
[tex]\[
Q3 = 10 + (10 - Q1)
\][/tex]
[tex]\[
Q3 = 20 - Q1
\][/tex]
4. Set Up the Equation Using IQR: We know that the IQR is 10, so:
[tex]\[
IQR = 10 = Q3 - Q1
\][/tex]
Substituting the expression for Q3 into this equation gives:
[tex]\[
10 = (20 - Q1) - Q1
\][/tex]
Simplifying this:
[tex]\[
10 = 20 - 2Q1
\][/tex]
5. Solve for Q1: Isolate Q1 on one side of the equation:
[tex]\[
10 = 20 - 2Q1
\][/tex]
[tex]\[
2Q1 = 20 - 10
\][/tex]
[tex]\[
2Q1 = 10
\][/tex]
[tex]\[
Q1 = \frac{10}{2}
\][/tex]
[tex]\[
Q1 = 5
\][/tex]
So, the First Quartile (Q1) is [tex]\(5.0\)[/tex].
