Question

independent practice
using the line provided, construct a box - and - whisker plot for the 12 scores below.
26, 32, 19, 65, 57, 16, 28, 42, 40, 21, 38, 10
determine the number of scores that lie above the 3rd quartile.
3

Explanation:

Step1: Order the data

First, we order the 12 scores from least to greatest: \(10, 16, 19, 21, 26, 28, 32, 38, 40, 42, 57, 65\)

Step2: Find the quartiles

The formula for the position of the \(n\)-th quartile (\(Q_n\)) is \(Q_n = \frac{n(n + 1)}{4}\) -th term (for \(n = 1, 2, 3\)). For \(n = 3\) (third quartile, \(Q_3\)):
The position of \(Q_3\) is \(\frac{3(12 + 1)}{4}=\frac{3\times13}{4}=9.75\) -th term. This means we take the 9th term plus 0.75 times the difference between the 10th and 9th terms.
The 9th term is \(40\), the 10th term is \(42\). So \(Q_3 = 40+0.75\times(42 - 40)=40 + 1.5 = 41.5\)
But another way to think about quartiles for a set with \(N\) data points (where \(N\) is divisible by 4 or not) is:
For \(N = 12\), the data is divided into 4 groups of 3 numbers each.
First group: \(10, 16, 19\) ( \(Q_1\) is the median of this group, but we care about \(Q_3\))
Second group: \(21, 26, 28\)
Third group: \(32, 38, 40\)
Fourth group: \(42, 57, 65\)
The third quartile (\(Q_3\)) is the median of the upper half (last 6 numbers: \(32, 38, 40, 42, 57, 65\)). The median of these 6 numbers is the average of the 3rd and 4th terms. The 3rd term is \(40\), the 4th term is \(42\), so \(Q_3=\frac{40 + 42}{2}=41\) (or using the position method we had \(41.5\), but in terms of counting how many numbers are above \(Q_3\), we can also use the fact that in a data set with \(N\) elements, the number of elements above \(Q_3\) is \(\frac{N}{4}\) when \(N\) is divisible by 4. Since \(N = 12\), \(\frac{12}{4}=3\)

Step3: Count the numbers above \(Q_3\)

Since the data is divided into 4 equal parts (each with \(12\div4 = 3\) numbers), the numbers above \(Q_3\) are the last 3 numbers in the ordered list. Looking at the ordered data: \(10, 16, 19, 21, 26, 28, 32, 38, 40, 42, 57, 65\), the last 3 numbers are \(42, 57, 65\), so there are 3 numbers above \(Q_3\)

Answer:

\(3\)