Question

11, 19, 16, 25, 22, 15, 13, 10, 17, 24, 21, 4, 3
minimum: _____________ maximum: _____________
$q_1$: ________ $q_2$: ______ $q_3$: ________
type out the 5 number data summary and clearly label each value

Explanation:

Step1: Sort the data

First, we sort the given data set: \( 3, 4, 10, 11, 13, 15, 16, 17, 19, 21, 22, 24, 25 \)

Step2: Find the minimum

The minimum value is the smallest number in the sorted data set. So, Minimum = \( 3 \)

Step3: Find the maximum

The maximum value is the largest number in the sorted data set. So, Maximum = \( 25 \)

Step4: Find the median (\( Q_2 \))

The number of data points \( n = 13 \) (odd). The median ( \( Q_2 \)) is the middle value, which is at position \( \frac{n + 1}{2}=\frac{13+ 1}{2}=7 \)th term. The 7th term in the sorted data is \( 16 \), so \( Q_2 = 16 \)

Step5: Find \( Q_1 \) (First Quartile)

\( Q_1 \) is the median of the lower half of the data. The lower half of the data (excluding the median) is \( 3, 4, 10, 11, 13, 15 \) ( \( n = 6 \), even). The median of this set is the average of the 3rd and 4th terms. 3rd term = \( 10 \), 4th term = \( 11 \). So, \( Q_1=\frac{10 + 11}{2}=\frac{21}{2}=10.5 \)

Step6: Find \( Q_3 \) (Third Quartile)

\( Q_3 \) is the median of the upper half of the data. The upper half of the data (excluding the median) is \( 17, 19, 21, 22, 24, 25 \) ( \( n = 6 \), even). The median of this set is the average of the 3rd and 4th terms. 3rd term = \( 21 \), 4th term = \( 22 \). So, \( Q_3=\frac{21+ 22}{2}=\frac{43}{2}=21.5 \)

Answer:

Minimum: \( 3 \)
Maximum: \( 25 \)
\( Q_1 \): \( 10.5 \)
\( Q_2 \): \( 16 \)
\( Q_3 \): \( 21.5 \)

The five - number summary is: Minimum = \( 3 \), \( Q_1 = 10.5 \), \( Q_2 = 16 \), \( Q_3 = 21.5 \), Maximum = \( 25 \)