Answer :
The lower quartile (Q1) is 149.5 pounds, the upper quartile (Q3) is 170 pounds, and the interquartile range is 20.5 pounds.
Determining Quartiles and Interquartile Range
The given data represent the weights in pounds of a sample of 25 police officers. Here are the steps to determine the requested statistical measures:
- Order the Data: First, we need to arrange the weights in ascending order: 134, 137, 140, 145, 146, 148, 151, 152, 154, 156, 156, 157, 162, 163, 164, 165, 168, 168, 169, 171, 172, 173, 174, 174, 177.
- Calculate Q1 (Lower Quartile): The lower quartile (Q1) is the median of the first half of the data (excluding the overall median if the number is odd).
- In this ordered set, the position of Q1 is (25 + 1) / 4 = 6.5th position.
- The value of Q1 is the average of the 6th and 7th values, (148 + 151) / 2 = 149.5.
- Calculate Q3 (Upper Quartile): The upper quartile (Q3) is the median of the second half of the data.
- The position of Q3 is 3 × (25 + 1) / 4 = 19.5th position.
- The value of Q3 is the average of the 19th and 20th values, (169 + 171) / 2 = 170.
- Find Interquartile Range (IQR): IQR is Q3 - Q1. IQR = 170 - 149.5 = 20.5.
Summary
The lower quartile (Q1) is 149.5 pounds, the upper quartile (Q3) is 170 pounds, and the interquartile range is 20.5 pounds.
Given:
The data values are:
164, 148, 137, 157, 173, 156, 177, 172, 169, 165, 145, 168, 163, 162, 174, 152, 156, 168, 154, 151, 174, 146, 134, 140, and 171.
To find;
a. Lower quartile.
b. Upper quartile.
c. Interquartile range.
Solution:
We have,
164, 148, 137, 157, 173, 156, 177, 172, 169, 165, 145, 168, 163, 162, 174, 152, 156, 168, 154, 151, 174, 146, 134, 140, 171.
Arrange the data values in ascending order.
134, 137, 140, 145, 146, 148, 151, 152, 154, 156, 156, 157, 162, 163, 164, 165, 168, 168, 169, 171, 172, 173, 174, 174, 177.
Divide the data values in two equal parts.
(134, 137, 140, 145, 146, 148, 151, 152, 154, 156, 156, 157), 162, (163, 164, 165, 168, 168, 169, 171, 172, 173, 174, 174, 177)
Divide each parentheses in two equal parts.
(134, 137, 140, 145, 146, 148), (151, 152, 154, 156, 156, 157), 162, (163, 164, 165, 168, 168, 169), (171, 172, 173, 174, 174, 177)
a. Location of lower quartile is:
[tex]Q_1=\dfrac{1}{4}(n+1)\text{th term}[/tex]
[tex]Q_1=\dfrac{1}{4}(25+1)\text{th term}[/tex]
[tex]Q_1=\dfrac{26}{4}\text{th term}[/tex]
[tex]Q_1=6.5\text{th term}[/tex]
The lower quartile of the weights is:
[tex]Q_1=\dfrac{148+151}{2}[/tex]
[tex]Q_1=\dfrac{299}{2}[/tex]
[tex]Q_1=149.5[/tex]
Therefore, the location of the lower quartile of the weights is between 6th term and the 7th term. The value of the lower quartile is 149.5.
b. Location of upper quartile is:
[tex]Q_3=\dfrac{3}{4}(n+1)\text{th term}[/tex]
[tex]Q_3=\dfrac{3}{4}(25+1)\text{th term}[/tex]
[tex]Q_3=\dfrac{3\cdot 26}{4}\text{th term}[/tex]
[tex]Q_3=19.5\text{th term}[/tex]
The upper quartile of the weights is:
[tex]Q_3=\dfrac{169+171}{2}[/tex]
[tex]Q_3=\dfrac{340}{2}[/tex]
[tex]Q_3=170[/tex]
Therefore, the location of the upper quartile of the weights is between 19th term and the 20th term. The value of the upper quartile is 170.
c. The interquartile range of the given data set is:
[tex]IQR=Q_3-Q_1[/tex]
[tex]IQR=170-149.5[/tex]
[tex]IQR=20.5[/tex]
Therefore, the interquartile range of the weights is 20.5.
