Question

the data set represents the number of rings each person in a room is wearing 0, 2, 4, 0, 2, 3, 2, 8, 6 what is the interquartile range of the data?
○ 2
○ 3
○ 4
○ 6

Explanation:

Step1: Order the data set

First, we order the given data set: \(0, 0, 2, 2, 2, 3, 4, 6, 8\)

Step2: Find the median (Q2)

The number of data points \(n = 9\), which is odd. The median is the middle value, at position \(\frac{n + 1}{2}=\frac{9+ 1}{2}=5\)th value. So the median (Q2) is \(2\).

Step3: Find Q1 (median of lower half)

The lower half of the data (values below Q2) is \(0, 0, 2, 2\). The number of values here is \(4\) (even). The median of this subset is the average of the 2nd and 3rd values. The 2nd value is \(0\) and the 3rd value is \(2\), so \(Q1=\frac{0 + 2}{2}=1\)? Wait, no, wait. Wait, the lower half for \(n = 9\) (odd) is the first \(\frac{n - 1}{2}=4\) values: \(0,0,2,2\). Wait, actually, when \(n\) is odd, the median is excluded from the lower and upper halves. So lower half is first 4 values: \(0,0,2,2\), upper half is last 4 values: \(3,4,6,8\). Wait, no, let's correct. For \(n = 9\), positions are 1:0, 2:0, 3:2, 4:2, 5:2 (median), 6:3, 7:4, 8:6, 9:8. So lower half is positions 1 - 4: \(0,0,2,2\), upper half is positions 6 - 9: \(3,4,6,8\). Now, Q1 is the median of lower half (4 values, even), so average of 2nd and 3rd in lower half. Lower half ordered: \(0,0,2,2\). 2nd value: \(0\), 3rd value: \(2\), so \(Q1=\frac{0 + 2}{2}=1\)? No, wait, no, maybe I made a mistake. Wait, no, the data set after ordering is \(0,0,2,2,2,3,4,6,8\). Let's use the formula for quartiles. Another method: for a data set with \(n\) elements, the first quartile \(Q1\) is at position \(\frac{n + 1}{4}\) when \(n\) is not a multiple of 4, or average of positions \(\frac{n}{4}\) and \(\frac{n}{4}+ 1\) when \(n\) is a multiple of 4. Here \(n = 9\), so \(\frac{9+ 1}{4}=2.5\). So Q1 is the average of the 2nd and 3rd values. 2nd value: \(0\), 3rd value: \(2\), so \(Q1=\frac{0 + 2}{2}=1\)? Wait, no, that can't be. Wait, maybe I messed up the ordering. Wait the original data is \(0,2,4,0,2,3,2,8,6\). Let's re - order: \(0,0,2,2,2,3,4,6,8\). Yes, that's correct. Now, Q3: position \(\frac{3(n + 1)}{4}=\frac{3\times10}{4}=7.5\). So Q3 is the average of 7th and 8th values. 7th value: \(4\), 8th value: \(6\), so \(Q3=\frac{4 + 6}{2}=5\). Then interquartile range (IQR) is \(Q3 - Q1\). Wait, but earlier calculation of Q1 was wrong. Wait, let's use the method of dividing the data into lower half and upper half excluding the median. The data is \(0,0,2,2,2,3,4,6,8\). Median (Q2) is the 5th value, which is \(2\). Lower half (values less than Q2? No, lower half is the first four values: \(0,0,2,2\) (since we exclude the median). Upper half is the last four values: \(3,4,6,8\). Now, median of lower half (Q1): the lower half has 4 values, so median is average of 2nd and 3rd. 2nd value in lower half: \(0\), 3rd value: \(2\), so \(Q1=\frac{0 + 2}{2}=1\). Median of upper half (Q3): upper half has 4 values, median is average of 2nd and 3rd. 2nd value in upper half: \(4\), 3rd value: \(6\), so \(Q3=\frac{4+6}{2}=5\). Then IQR = \(Q3 - Q1=5 - 1 = 4\)? Wait, but that doesn't match. Wait, maybe another approach. Let's use the formula for quartiles in a data set with \(n\) elements. For \(n = 9\), the positions are:

- \(Q1\) position: \(\frac{n + 1}{4}=\frac{9 + 1}{4}=2.5\), so \(Q1\) is the value at 2.5th position, which is the average of the 2nd and 3rd values. 2nd value: \(0\), 3rd value: \(2\), so \(Q1 = 1\)
- \(Q3\) position: \(\frac{3(n + 1)}{4}=\frac{3\times10}{4}=7.5\), so \(Q3\) is the average of the 7th and 8th values. 7th value: \(4\), 8th value: \(6\), so \(Q3 = 5\)
- IQR=Q3 - Q1 = 5 - 1=4? Wait, but let's check again. Wait, maybe I made a mistake in the data set. Wait the original data is \(0,2,4,0,2,3,2,8,6\). Let's count the number of elements: 9 elements. Ordered: \(0,0,2,2,2,3,4,6,8\). Let's list the positions:

1: 0

2: 0

3: 2

4: 2

5: 2 (median)

6: 3

7: 4

8: 6

9: 8

Now, lower quartile (Q1) is the median of the first 4 numbers (positions 1 - 4: 0,0,2,2). The median of 0,0,2,2 is \(\frac{0 + 2}{2}=1\)

Upper quartile (Q3) is the median of the last 4 numbers (positions 6 - 9: 3,4,6,8). The median of 3,4,6,8 is \(\frac{4 + 6}{2}=5\)

Then interquartile range is \(Q3 - Q1=5 - 1 = 4\)

Wait, but let's check with another method. Some sources define quartiles as follows: for a data set with \(n\) observations, ordered from smallest to largest.

- If \(n\) is odd, the median is the middle value, and we split the data into lower half (values below median) and upper half (values above median).

- \(Q1\) is the median of the lower half, \(Q3\) is the median of the upper half.

In our case, lower half (values below median \(2\)): \(0,0,2,2\) (4 values), median of lower half: \(\frac{0 + 2}{2}=1\)

Upper half (values above median \(2\)): \(3,4,6,8\) (4 values), median of upper half: \(\frac{4+6}{2}=5\)

IQR = \(5 - 1=4\)

Answer:

4