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= 10. kate kozaks 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 1.6 3.7 6 11.7 14.1 16.4 20.4 25.1 28 28.8 median = q1 = q3 =

Explanation:

Step1: Find the median

Since \( n = 10 \) (even), the median is the average of the 5th and 6th values. The 5th value is \( 14.1 \), the 6th is \( 16.4 \). So median \(=\frac{14.1 + 16.4}{2}=\frac{30.5}{2}=15.25\).

Step2: Find Q1

First half of data (excluding median): values are \( 1.6, 3.7, 6, 11.7, 14.1 \)? Wait, no, when \( n = 10 \), first half is first 5 values? Wait, no, the data is 10 values: positions 1 - 10. Median is between 5th and 6th. So first half (for Q1) is first 5 values? Wait, no, the algorithm says: after median, Q1 is median of 1st half not including median. Wait, \( n = 10 \), so the data is split into two halves: first 5 (positions 1 - 5: \( 1.6, 3.7, 6, 11.7, 14.1 \)) and second 5 (positions 6 - 10: \( 16.4, 20.4, 25.1, 28, 28.8 \))? Wait, no, when \( n \) is even, the median is between the 5th and 6th. So the first half (for Q1) is the first 5 values (positions 1 - 5), and the second half (for Q3) is the last 5 values (positions 6 - 10). Wait, but the algorithm says "1st half of the data, not including the median value". Wait, the data has 10 values, median is between 5th and 6th. So the first half (for Q1) is the first 5 values (since median is between 5 and 6, so first half is 1 - 5, second half 6 - 10). So for Q1, we take the median of first 5 values. First 5 values: \( 1.6, 3.7, 6, 11.7, 14.1 \). The median of these 5 (odd number) is the 3rd value, which is \( 6 \). Wait, no, wait: 5 values, median is the 3rd. So \( 1.6, 3.7, 6, 11.7, 14.1 \): sorted, so median is 3rd value, which is \( 6 \). Wait, but wait, maybe I made a mistake. Wait the data is: 1.6, 3.7, 6, 11.7, 14.1, 16.4, 20.4, 25.1, 28, 28.8. So first half (for Q1) is the first 5 values: 1.6, 3.7, 6, 11.7, 14.1. The median of these 5 is the 3rd value, which is 6. Wait, but let's check again. Wait, the algorithm: "after computing the median, Q1 is the median of the 1st half of the data, not including the median value". So the 1st half: when \( n = 10 \), the data is divided into two groups: lower half (values below median) and upper half (values above median). Wait, no, the median is 15.25, so the lower half is all values less than 15.25: \( 1.6, 3.7, 6, 11.7, 14.1 \) (5 values), upper half: \( 16.4, 20.4, 25.1, 28, 28.8 \) (5 values). So Q1 is the median of lower half: 5 values, median is 3rd value: \( 6 \).

Step3: Find Q3

Q3 is the median of the upper half (values above median: \( 16.4, 20.4, 25.1, 28, 28.8 \)). This is 5 values, median is 3rd value: \( 25.1 \).

Answer:

Median = \( 15.25 \), Q1 = \( 6 \), Q3 = \( 25.1 \)