Question

people at a library were asked how many states they had visited. below is a stemplot of all of the data except for one value. which of the following statements is true? if the missing data value were 32, the iqr would change substantially. if the missing data value were 32, the range would change substantially. if the missing data value were 50, the iqr of the data would change substantially. if the missing data value were 50, the range of the data would change substantially. number of states visited 0 8 9 9 9 1 0 1 2 2 2 4 5 6 6 8 8 2 1 2 7 7 7 3 1 key: 2|7 = 27 states

Explanation:

Step1: Recall the definitions of range and IQR

• Range: The range is the difference between the maximum and minimum values in a data set (\(Range = Max - Min\)).
• IQR (Inter - Quartile Range): \(IQR=Q_{3}-Q_{1}\), where \(Q_{1}\) is the first quartile (the value that separates the lowest \(25\%\) of the data from the rest) and \(Q_{3}\) is the third quartile (the value that separates the lowest \(75\%\) of the data from the highest \(25\%\)). Quartiles are based on the position of data values when the data is ordered.

Step2: Analyze the effect of adding 32

• Current data: The minimum value (\(Min\)) is \(8\), and the current maximum value (\(Max\)) (excluding the missing value) is \(31\). If the missing value is \(32\), the new \(Max = 32\). The change in the range is \(32 - 8-(31 - 8)=1\).
• For the IQR: When calculating quartiles, adding a value close to the existing upper - end values (since \(32\) is close to \(31\)) does not change the position of \(Q_{1}\) and \(Q_{3}\) (because quartiles depend on the relative position of data points. If the data set size \(n\) is considered, and we use the formula \(n + 1\) for quartile positions (\(Q_{1}\) at \(\frac{n + 1}{4}\)th position and \(Q_{3}\) at \(\frac{3(n + 1)}{4}\)th position). With \(n\) (original non - missing data points: \(4+12 + 5+1=22\) data points, after adding one more \(n = 23\)), the positions of \(Q_{1}\) and \(Q_{3}\) are based on ordered data. Since \(32\) is just one value above \(31\), it does not shift the quartile - defining data points.

Step3: Analyze the effect of adding 50

• Range: The current \(Min = 8\). If the missing value is \(50\), the new \(Max=50\). The change in the range is \(50 - 8-(31 - 8)=19\).
• IQR: When calculating quartiles (\(Q_{1}\) and \(Q_{3}\)), since \(50\) is an outlier (a value that is much larger than the other values). But quartiles are resistant to outliers. The first quartile \(Q_{1}\) is based on the lower half of the data and \(Q_{3}\) is based on the upper half of the data. Adding \(50\) does not change the values that define \(Q_{1}\) and \(Q_{3}\) (because when we order the data, the values that determine the quartiles (the values in the lower \(25\%\) and upper \(25\%\) boundaries) are not affected by a single large outlier. For example, if \(n = 23\) (after adding the missing value), \(Q_{1}\) is at the \(\frac{23 + 1}{4}=6\)th ordered value and \(Q_{3}\) is at the \(\frac{3(23 + 1)}{4}=18\)th ordered value. A single large value at the end of the ordered data set does not change these positions for \(Q_{1}\) and \(Q_{3}\)).

Answer:

If the missing data value were \(50\), the range of the data would change substantially.