Question

1. what are the lower, middle, and upper quartiles of this data? 23, 15, 22, 15, 23, 15, 13, 21, 14 lower: 14.5, middle: 15, upper: 22.5 lower: 14, middle: 15, upper: 23 lower: 13, middle: 15, upper: 23 lower: 15, middle: 15, upper: 22

Explanation:

Step1: Order the data

First, we order the data set: \(13, 14, 15, 15, 15, 21, 22, 23, 23\) (wait, original data has 9 elements? Wait, original data: 23,15,22,15,23,15,13,21,14. Let's count: 9 numbers. So ordered: \(13, 14, 15, 15, 15, 21, 22, 23, 23\).

Step2: Find the median (middle quartile, Q2)

The median (Q2) is the middle value. For \(n = 9\) (odd), the median is the \(\frac{9 + 1}{2}=5\)-th term. The 5th term is \(15\), so Q2 (middle) is \(15\).

Step3: Find lower quartile (Q1)

The lower half is the first 4 terms (since median is at 5th, lower half is terms 1 - 4: \(13, 14, 15, 15\)). The median of the lower half: for \(n = 4\) (even), median is \(\frac{14 + 15}{2}=\frac{29}{2}=14.5\). So Q1 (lower) is \(14.5\).

Step4: Find upper quartile (Q3)

The upper half is terms 6 - 9: \(21, 22, 23, 23\). The median of the upper half: for \(n = 4\) (even), median is \(\frac{22 + 23}{2}=\frac{45}{2}=22.5\). So Q3 (upper) is \(22.5\).

Answer:

lower: 14.5, middle: 15, upper: 22.5 (the first option)