Question

which statement best describes the interquartile range of this set of weights? {120 lb, 115 lb, 135 lb, 105 lb, 80 lb, 160 lb} the difference between the first and sixth elements of the set after it has been ordered from least to greatest the difference between the first and sixth elements of the set as it is written above the difference between the second and fifth elements of the set after it has been ordered from least to greatest the difference between the second and fifth elements of the set as it is written above

Explanation:

Step1: Sort the data set

First, we sort the data set \(\{80,105,115,120,135,160\}\) from least to greatest.

Step2: Find the first quartile (\(Q_1\)) and the third quartile (\(Q_3\))

The inter - quartile range (IQR) is \(Q_3 - Q_1\). For a set with \(n = 6\) data points, the first quartile (\(Q_1\)) is the value of the \(\frac{n + 1}{4}\)th ordered data point. \(\frac{6+1}{4}=1.75\)th data point. Using linear interpolation, \(Q_1=105+(115 - 105)\times0.75 = 112.5\). The third quartile (\(Q_3\)) is the value of the \(\frac{3(n + 1)}{4}\)th ordered data point. \(\frac{3\times(6 + 1)}{4}=5.25\)th data point. Using linear interpolation, \(Q_3=135+(160 - 135)\times0.25=141.25\). The inter - quartile range \(IQR=Q_3 - Q_1=141.25-112.5 = 28.75\).

Another way: The inter - quartile range is the difference between the third quartile and the first quartile. When the data is ordered from least to greatest, the first quartile (\(Q_1\)) is the median of the lower half of the data and the third quartile (\(Q_3\)) is the median of the upper half of the data.
The lower half of the ordered data \(\{80,105,115\}\) has a median \(Q_1 = 105\) (if we use the second - element - based on some definitions for small \(n\)). The upper half of the ordered data \(\{120,135,160\}\) has a median \(Q_3=135\). Then \(IQR = 135-105=30\) (using the simple non - interpolation method for small \(n\) data sets in some basic statistics courses).

The inter - quartile range is the difference between the third quartile and the first quartile. When the data is ordered from least to greatest, the first quartile (\(Q_1\)) is the value such that 25% of the data is less than it and the third quartile (\(Q_3\)) is the value such that 75% of the data is less than it.

The inter - quartile range is the difference between the third and first quartiles. When the data is ordered from least to greatest, the first quartile (\(Q_1\)) is the median of the first three elements and the third quartile (\(Q_3\)) is the median of the last three elements.

The data set \(\{80,105,115,120,135,160\}\) ordered from least to greatest. The first three elements are \(\{80,105,115\}\), median \(Q_1 = 105\). The last three elements are \(\{120,135,160\}\), median \(Q_3=135\). \(IQR=Q_3 - Q_1=135 - 105=30\)

Answer:

The inter - quartile range is the difference between the third and first quartiles. When the data is ordered from least to greatest, the first quartile (\(Q_1\)) is the median of the first three elements and the third quartile (\(Q_3\)) is the median of the last three elements. So the inter - quartile range is the difference between the third and first quartiles after the data has been ordered from least to greatest.