Question

The diameter at breast height (DBH) of 20 trees is as follows:

\[ 1,\ 3,\ 5,\ 5,\ 7,\ 8,\ 8,\ 8,\ 8,\ 8,\ 8,\ 9,\ 9,\ 9,\ 9,\ 9,\ 10,\ 10,\ 10,\ 10 \] 

What are the values of the first quartile \((Q_1)\), second quartile \((Q_2)\), and third quartile \((Q_3)\)?

1. \(Q_1 = 7.5,\ Q_2 = 8,\ Q_3 = 8\)

2. \(Q_1 = 7.5,\ Q_2 = 8,\ Q_3 = 9\)

3. \(Q_1 = 8,\ Q_2 = 9,\ Q_3 = 10\)

4. \(Q_1 = 8,\ Q_2 = 8.5,\ Q_3 = 9\)

• A. \(Q_1 = 7.5\), \(Q_2 = 8\) and \(Q_3 = 8\)
• B. \(Q_1 = 7.5\), \(Q_2 = 8\) and \(Q_3 = 9\)
• C. \(Q_1 = 8\), \(Q_2 = 9\) and \(Q_3 = 10\)
• D. \(Q_1 = 8\), \(Q_2 = 8.5\) and \(Q_3 = 9\)

Hint:

For small even datasets, dividing the data into lower and upper halves and finding the median of each half (known as the method of hinges) is the most efficient and reliable way to compute quartiles.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept

This problem requires finding the three quartiles, \(Q_1\), \(Q_2\), and \(Q_3\), of an ordered dataset containing \(n = 20\) observations.

Key Formula or Approach:

Quartiles divide an ordered dataset into four equal parts. For an even number of observations:

• \(Q_2\) (Median): Average of the \(10^{\text{th}}\) and \(11^{\text{th}}\) observations.
• \(Q_1\) (First Quartile): Median of the first 10 observations.
• \(Q_3\) (Third Quartile): Median of the last 10 observations.

Step 2: Detailed Calculation

The ordered data with their positions is:

\[ \begin{array}{c|cccccccccccccccccccc} \text{Index} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ \hline \text{Value} & 1 & 3 & 5 & 5 & 7 & 8 & 8 & 8 & 8 & 8 & 8 & 9 & 9 & 9 & 9 & 9 & 10 & 10 & 10 & 10 \end{array} \]

Finding \(Q_2\) (Median):

Since \(n = 20\) is even, the median is the average of the \(10^{\text{th}}\) and \(11^{\text{th}}\) observations.

\[ Q_2=\frac{X_{10}+X_{11}}{2} =\frac{8+8}{2} =8 \]

Finding \(Q_1\) (First Quartile):

The lower half consists of the first 10 observations:

\[ 1,\;3,\;5,\;5,\;7,\;8,\;8,\;8,\;8,\;8 \]

The median of these 10 observations is the average of the \(5^{\text{th}}\) and \(6^{\text{th}}\) values.

\[ Q_1=\frac{7+8}{2}=7.5 \]

Finding \(Q_3\) (Third Quartile):

The upper half consists of the last 10 observations:

\[ 8,\;9,\;9,\;9,\;9,\;9,\;10,\;10,\;10,\;10 \]

The median of these observations is the average of the \(5^{\text{th}}\) and \(6^{\text{th}}\) values of this half.

\[ Q_3=\frac{9+9}{2}=9 \]

Step 3: Final Answer

Therefore,

\[ Q_1=7.5,\qquad Q_2=8,\qquad Q_3=9 \]

Hence, the correct answer is

\[ \boxed{\text{Option (B)}} \]