Question

question #11
find the interquartile range for the following set of data:
15 13 17 18 13
13 16 22
22 13 14 17 22
21 15 23
3
4.5
8
2

Explanation:

Step1: Sort the data

Sort the data set in ascending order: \(13,13,13,13,14,15,15,16,17,17,18,21,22,22,22,23\)

Step2: Find the median (Q2)

Since there are \(n = 16\) data points. The median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+ 1)\)th values. \(\frac{16}{2}=8\) and \(\frac{16}{2}+1 = 9\). The 8th value is \(16\) and the 9th value is \(17\). So \(Q2=\frac{16 + 17}{2}=16.5\)

Step3: Find Q1

The lower - half of the data is \(13,13,13,13,14,15,15,16\). Since \(n_1=8\) (number of data points in lower - half), \(Q1\) is the average of the 4th and 5th values. The 4th value is \(13\) and the 5th value is \(14\). So \(Q1=\frac{13+14}{2}=13.5\)

Step4: Find Q3

The upper - half of the data is \(17,18,21,22,22,22,23\). Since \(n_2 = 8\) (number of data points in upper - half), \(Q3\) is the average of the 4th and 5th values. The 4th value is \(22\) and the 5th value is \(22\). So \(Q3 = 22\)

Step5: Calculate the Inter - quartile Range (IQR)

\(IQR=Q3 - Q1\)
\(IQR=22-13.5 = 8.5\)

Wait, there is a mistake above. Let's re - calculate:

Sort the data: \(13,13,13,13,14,15,15,16,17,17,18,21,22,22,22,23\)

For \(n = 16\) (even number of data points)

Median position: \(\frac{n}{2}=8\) and \(\frac{n}{2}+1 = 9\). The data set is divided into two halves.

Lower half: \(13,13,13,13,14,15,15,16\)

\(Q1\): For \(n_1=8\) (lower half), \(Q1\) is the 4th value (since \(\frac{n_1 + 1}{2}=4.5\), average of 4th and 5th. Wait, no, for \(n_1 = 8\) (even), \(Q1=\frac{4^{th}+5^{th}}{2}\). \(4^{th}=13\), \(5^{th}=14\), \(Q1 = 13.5\)

Upper half: \(17,18,21,22,22,22,23\)

For \(n_2=8\) (upper half), \(Q3\) is the 6th value (since \(\frac{n_2+1}{2}=4.5\), average of 4th and 5th. Wait, no, for \(n_2 = 8\) (even), \(Q3=\frac{6^{th}+7^{th}}{2}\). \(6^{th}=22\), \(7^{th}=22\), \(Q3 = 22\)

\(IQR=Q3 - Q1=22-13.5 = 8.5\)

Wait, another approach:

Sort the data: \(13,13,13,13,14,15,15,16,17,17,18,21,22,22,22,23\)

Number of data points \(n = 16\)

\(Q1\): The median of the first 8 data points. The first 8 data points: \(13,13,13,13,14,15,15,16\). Median of these (since \(n = 8\)) is \(\frac{13 + 14}{2}=13.5\)

\(Q3\): The median of the last 8 data points. The last 8 data points: \(17,18,21,22,22,22,23\). Median of these (since \(n = 8\)) is \(\frac{22+22}{2}=22\)

\(IQR=Q3 - Q1=22 - 13.5=8.5\)

But if we consider the formula \(IQR = Q3-Q1\) with another definition (some textbooks define \(Q1\) as the \(\frac{n + 1}{4}\)th value and \(Q3\) as the \(\frac{3(n + 1)}{4}\)th value for \(n\) data points)

For \(n = 16\), \(\frac{n+1}{4}=\frac{17}{4}=4.25\), \(Q1\) is \(13+0.25\times(14 - 13)=13.25\)

\(\frac{3(n + 1)}{4}=\frac{51}{4}=12.75\), \(Q3\) is \(21+0.75\times(22 - 21)=21.75\)

\(IQR=21.75-13.25 = 8.5\)

If there is a miscalculation in the problem's options, and assuming we made a wrong data - split (if we consider \(n = 15\) by mistake, but no, \(n = 16\))

Wait, re - check the data:

Original data:
Row 1: \(15,13,17,18,13\)
Row 2: \(13,16,22\)
Row 3: \(22,13,14,17,22\)
Row 4: \(21,15,23\)

Count the data points: \(5 + 3+5+3=16\)

Another way (Tukey's method):

Sort: \(13,13,13,13,14,15,15,16,17,17,18,21,22,22,22,23\)

\(Q1\): median of \(13,13,13,13,14,15,15,16\) is \(\frac{13 + 14}{2}=13.5\)

\(Q3\): median of \(17,17,18,21,22,22,22,23\) is \(\frac{21+22}{2}=21.5\)

\(IQR=21.5-13.5 = 8\)

Answer:

8