Question Type 5: Constructing box and whisker plots using cumulative frequency graphs or tables Exercises

In a box plot the maximum is 36 and the range of the top 25% of data (from $Q_3$ to the maximum) is 12 units. Find the value of the upper quartile $Q_3$.

A box-and-whisker plot has a lower whisker at 10, lower quartile $Q_1=15$, median $=20$, upper quartile $Q_3=30$, and upper whisker at 40. Find: (a) The percentage of data between $Q_1$ and $Q_3$. (b) The range of the middle 50% of the data.

The box-and-whisker plot below shows the lower whisker at 4, the box from 7 to 17, the median at 12, and the upper whisker at 25. Determine the interquartile range (IQR) and state whether there are any outliers using the $1.5\times\mathrm{IQR}$ rule.

Given the cumulative frequency table below for test scores, find the lower quartile $Q_1$, the median, the upper quartile $Q_3$, and state the five-number summary.

Two datasets A and B have box plots with the following five-number summaries:

A: min=10, $Q_1=20$, median=30, $Q_3=40$, max=50

B: min=5, $Q_1=15$, median=25, $Q_3=35$, max=45

Compare the central tendency and variability of A and B.

The cumulative frequency graph of weekly study hours for 40 students is plotted by joining the points $(5,4)$, $(10,12)$, $(15,20)$, $(20,30)$, $(25,36)$ and $(30,40)$. Use this graph to estimate $Q_1$, the median and $Q_3$.

Two datasets X and Y have box plots with these statistics:

X: range = 50, IQR = 25

Y: range = 40, IQR = 30

Which dataset shows greater overall variability, and which shows greater middle-50% variability? Justify your answer.

The cumulative frequency curve for a dataset of 20 measurements passes through the points $(2,4)$, $(4,9)$, $(6,15)$, $(8,18)$ and $(10,20)$. Estimate the lower quartile $Q_1$, the median and the upper quartile $Q_3$.

Dataset C has five-number summary $(5,15,20,25,80)$ and dataset D has $(5,15,20,25,50)$. Compare the skewness and presence of extreme values in C and D.

A grouped dataset of weights of 50 apples is given in the cumulative frequency table below. Estimate $Q_1$, the median and $Q_3$ using linear interpolation.

Box plot A has whiskers from 2 to 12 and box from 4 to 10 (median at 7). Box plot B has whiskers from 1 to 14 and box from 5 to 9 (median at 7). Compare the skewness and the presence of outliers of A and B.