Question

the measure of position called the midquartile of a data set is found using the formula \\(\frac{q_1 + q_3}{2}\\). find the midquartile of the given data set. 24 37 46 34 35 41 40 26 32 22 38 45. the midquartile is \\(\square\\). (type an integer or a decimal. do not round.)

Explanation:

Step1: Order the data set

First, we order the given data set: \(22, 24, 26, 32, 34, 35, 37, 38, 40, 41, 45, 46\)

Step2: Find the median position

The number of data points \(n = 12\). The median is the average of the 6th and 7th values. But we need \(Q_1\) (first quartile) and \(Q_3\) (third quartile). For \(n = 12\), the first quartile \(Q_1\) is the median of the first half of the data (first 6 values: \(22, 24, 26, 32, 34, 35\)) and \(Q_3\) is the median of the second half of the data (last 6 values: \(37, 38, 40, 41, 45, 46\)).

Step3: Calculate \(Q_1\)

For the first half (\(22, 24, 26, 32, 34, 35\)), the median ( \(Q_1\)) is the average of the 3rd and 4th values. So \(Q_1=\frac{26 + 32}{2}=\frac{58}{2}=29\)

Step4: Calculate \(Q_3\)

For the second half (\(37, 38, 40, 41, 45, 46\)), the median ( \(Q_3\)) is the average of the 3rd and 4th values. So \(Q_3=\frac{40+41}{2}=\frac{81}{2} = 40.5\)

Step5: Calculate the midquartile

Using the formula \(\frac{Q_1 + Q_3}{2}\), we substitute \(Q_1 = 29\) and \(Q_3=40.5\). So \(\frac{29+40.5}{2}=\frac{69.5}{2}=34.75\)

Answer:

\(34.75\)