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

Question 1

Skill question

The following cumulative frequency table displays the test scores of a group of students.

$Score Cum. Freq 4045012601970268030$

Find the lower quartile $Q_{1}$ , the median, and the upper quartile $Q_{3}$ . Hence, state the five-number summary for these test scores.

[5]

Question 2

Skill question

The following box-and-whisker plot displays the distribution of a dataset.

Determine the interquartile range ( $IQR$ ) and state whether there are any outliers using the $1.5 \times IQR$ rule.

[5]

Question 3

Skill question

Calculations involving linear interpolation for quartiles and the median from a grouped frequency distribution.

A grouped dataset of weights of 50 apples is given in the cumulative frequency table below. Estimate the lower quartile ( $Q_{1}$ ), the median and the upper quartile ( $Q_{3}$ ) using linear interpolation.

$Weight (g) Cumulative Frequency 80 - 100 5 100 - 120 15 120 - 140 30 140 - 160 45 160 - 180 50$

[6]

Question 4

Skill question

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 $Q_{3}$ .

[5]

Question 5

Skill question

In a box plot, the maximum value is 36 and the range of the top 25% of the data (from the upper quartile $Q_{3}$ to the maximum) is 12 units. Find the value of $Q_{3}$ .

[2]

Question 6

Skill question

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.

1.

Find the percentage of data between $Q_{1}$ and $Q_{3}$ .

[1]

2.

Find the range of the middle $50%$ of the data.

[2]

Question 7

Skill question

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

[4]

Question 8

Skill question

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 $B$ . [4 marks]

[4]

Question 9

Skill question

Comparison of two datasets based on five-number summaries, evaluating skewness, range, and outlier presence.

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 $D$ .

[4]

Question 10

Skill question

Statistics — Measures of Dispersion

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. [3 marks]

[3]

Question 11

Skill question

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 $Q_{3}$ .

[4]

IB Mathematics Applications & Interpretation Question Type 5: Constructing Box and Whisker Plots Using Cumulative Frequency Graphs or Tables Exercises