Question 1
The following data shows the times (minutes) taken by 12 students to complete a puzzle:
\(8,\;12,\;15,\;9,\;21,\;14,\;18,\;11,\;25,\;10,\;16,\;13\)
(a)Find the median, lower quartile (\(Q_1\)), and upper quartile (\(Q_3\)).
(b)Calculate the IQR.
(c)Identify any outliers using the rule: outlier if \(Q_3+1.5\times\text{IQR}\).
(a)
Sort: 8,9,10,11,12,13,14,15,16,18,21,25
Median \(=\tfrac{13+14}{2}=13.5\); \(Q_1=\tfrac{10+11}{2}=10.5\); \(Q_3=\tfrac{16+18}{2}=17\)
Answer
\(Q_1=10.5,\;\text{Median}=13.5,\;Q_3=17\)
(c)
Bounds
Lower: \(10.5-9.75=0.75\); Upper: \(17+9.75=26.75\)
Answer
No outliers (all values within bounds)
Question 2
The cumulative frequency table below shows the marks of 80 students:
\[\begin{array}{c|c}\text{Mark}\leq&\text{Cumulative Freq}\\hline20&5\30&14\40&30\50&54\60&70\70&78\80&80\end{array}\]
(a)Estimate the median and quartiles from a cumulative frequency graph.
(b)Draw a box-and-whisker plot.
(a)
Read off at \(n/2=40,\;n/4=20,\;3n/4=60\) (linear interpolation between table points)
\(Q_1\): between 30 (cf14) and 40 (cf30): \(30+\tfrac{20-14}{30-14}(10)=33.75\). Median: between 40 (cf30) and 50 (cf54): \(40+\tfrac{40-30}{54-30}(10)\approx44.2\). \(Q_3\): between 50 (cf54) and 60 (cf70): \(50+\tfrac{60-54}{70-54}(10)=53.75\)
Answer
\(Q_1\approx33.75,\;\text{Median}\approx44.2,\;Q_3\approx53.75\)
(b)
Box plot
Min \(=20\), \(Q_1\approx33.75\), Med \(\approx44.2\), \(Q_3\approx53.75\), Max \(=80\)
Question 3
Two classes sit the same test. Their results are summarised:
\[\begin{array}{l|ccccc}&\text{Min}&Q_1&\text{Med}&Q_3&\text{Max}\\hline\text{Class A}&42&55&64&72&90\text{Class B}&38&48&60&78&95\end{array}\]
(a)Compare the median scores.
(b)Compare the spread using the IQR.
(c)Which class performed more consistently? Justify.
(a)
Answer
Class A median (64) is higher than Class B (60), suggesting A performed better on average.
(b)
IQRs
Class A: \(72-55=17\); Class B: \(78-48=30\)
Answer
Class B has a larger IQR — more spread out.
(c)
Answer
Class A — smaller IQR indicates more consistent performance.