Question

find the median, first quartile (q1) and the third quartile (q3) for the data shown, which is already sorted. the number of data is n= 8. kate kozak’s algorithm: sort the data and compute the median. when n is an odd number, the median is the center value. when n is an even number, the median is the center value. after computing the median, q1 is the median of the 1st half of the data, not including the median value. q3 is the median of the 2nd half of the data, not including the median value. do not round your answers. x 1 4 9 11 13 16 16 24 median = q1 = q3 =

Explanation:

Step1: Find the median

The data set is \(1, 4, 9, 11, 13, 16, 16, 24\) with \(n = 8\) (even). The median is the average of the \(\frac{n}{2}\)-th and \((\frac{n}{2}+1)\)-th values. So, \(\frac{8}{2}=4\)-th value is \(11\) and \((\frac{8}{2}+1)=5\)-th value is \(13\). Median \(=\frac{11 + 13}{2}=\frac{24}{2}=12\).

Step2: Find Q1

The first half of the data (excluding the median values) is \(1, 4, 9, 11\) (wait, no: when \(n = 8\), the first half is the first 4 values? Wait, no, the algorithm says "1st half of the data, not including the median value". Wait, the data is 8 values. The median is between the 4th and 5th. So the first half (excluding median) is the first 4 values? Wait, no, the data is \(1, 4, 9, 11, 13, 16, 16, 24\). The median is between 11 (4th) and 13 (5th). So the first half (excluding median) is the first 4 values: \(1, 4, 9, 11\)? Wait, no, the algorithm says "after computing the median, Q1 is the median of the 1st half of the data, not including the median value". Wait, the data has 8 values. The median is the average of 4th and 5th. So the first half (before the median) is the first 4 values: positions 1 - 4: \(1, 4, 9, 11\). The median of these 4 values: since \(n = 4\) (even), it's the average of 2nd and 3rd values. 2nd value is \(4\), 3rd is \(9\). So Q1 \(=\frac{4 + 9}{2}=\frac{13}{2}=6.5\)? Wait, no, wait the algorithm: "Sort the data and compute the median. When n is even, the median is the center value (average of two). After computing the median, Q1 is the median of the 1st half of the data, not including the median value. Q3 is the median of the 2nd half of the data, not including the median value." Wait, the data is \(1, 4, 9, 11, 13, 16, 16, 24\). The median is \(12\) (average of 11 and 13). The first half (excluding median) is the data before the median position. Wait, the median is between the 4th and 5th elements. So the first half (excluding median) is the first 4 elements: \(1, 4, 9, 11\)? Wait, no, the first half of the original data (8 elements) is the first 4 elements, and the second half is the last 4 elements? Wait, the algorithm says "1st half of the data, not including the median value". Wait, maybe the first half is the first \(\frac{n}{2}\) values. Wait, \(n = 8\), so first half is 4 values: \(1, 4, 9, 11\). The median of these 4 values: since \(n = 4\) (even), it's the average of the 2nd and 3rd values. 2nd value is \(4\), 3rd is \(9\). So Q1 \(=\frac{4 + 9}{2}=6.5\). Wait, but let's check again. Wait, the data is sorted: \(1, 4, 9, 11, 13, 16, 16, 24\). The median is between the 4th (11) and 5th (13) values. So the first half (excluding median) is the first 4 values: \(1, 4, 9, 11\). The median of these 4: average of 2nd and 3rd: \(\frac{4 + 9}{2}=6.5\).

Step3: Find Q3

The second half of the data (excluding the median value) is \(13, 16, 16, 24\) (the last 4 values: positions 5 - 8, but excluding the median? Wait, the median is between 4th and 5th, so the second half is the last 4 values: \(13, 16, 16, 24\). The median of these 4 values: since \(n = 4\) (even), it's the average of 2nd and 3rd values. 2nd value is \(16\), 3rd is \(16\). So Q3 \(=\frac{16 + 16}{2}=16\).

Answer:

Median \(= 12\), \(Q1 = 6.5\), \(Q3 = 16\)