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Question 1

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

\begin{array}{|c|c|c|c|c|c|} \hline \text{Score} & 40 & 50 & 60 & 70 & 80 \\ \hline \text{Cum. Freq} & 4 & 12 & 19 & 26 & 30 \\ \hline \end{array}

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

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

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

[5]

Question 3

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.

\begin{array}{|l|c|c|c|c|c|} \hline \text{Weight (g)} & 80\text{--}100 & 100\text{--}120 & 120\text{--}140 & 140\text{--}160 & 160\text{--}180 \\ \hline \text{Cumulative Frequency} & 5 & 15 & 30 & 45 & 50 \\ \hline \end{array}

[6]

Question 4

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

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

A box-and-whisker plot has a lower whisker at 10, lower quartile $Q_1 = 15$ , $\text{median} = 20$ , 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

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

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

$A: \text{min}=10, Q_1=20, \text{median}=30, Q_3=40, \text{max}=50$

$B: \text{min}=5, Q_1=15, \text{median}=25, Q_3=35, \text{max}=45$

Compare the central tendency and variability of $A$ and $B$ . [4 marks]

[4]

Question 9

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

Statistics — Measures of Dispersion

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

$X$ : $\text{range} = 50$ , $\text{IQR} = 25$

$Y$ : $\text{range} = 40$ , $\text{IQR} = 30$

Which dataset shows greater overall variability, and which shows greater middle-50% variability? Justify your answer. [3 marks]

[3]

Question 11

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]

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