SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Statistics, STD2 S1 2024 GEN1* 6

More than 11 000 athletes from more than 200 countries competed in the Tokyo Summer Olympic Games.

An analysis of the number of athletes per country produced the following five-number summary.

\begin{array}{|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \textbf{Minimum} \rule[-1ex]{0pt}{0pt}& \textbf{First quartile } & \textbf{Median } & \textbf{Third quartile} & \textbf{Maximum } \\
\hline
\rule{0pt}{2.5ex} 2 \rule[-1ex]{0pt}{0pt}& 5 & 11 & 48 & 613 \\
\hline
\end{array}

Find the smallest number of athletes per country that would display as an outlier on a boxplot of this dataset.   (2 marks)

Show Answers Only

\(C\)

Show Worked Solution

\(IQR=48-5=43\)

\(\text{Upper boundary}\) \(=Q_3+1.5\times IQR\)
  \(=48+1.5\times 43\)
  \(=112.5\)

 
\(\therefore\ \text{Smallest number of athletes to show as an outlier = 113.}\)

Statistics, STD2 S1 2016 VCE-G 2*

A weather station records daily maximum temperatures

The five-number summary for the distribution of maximum temperatures for the month of February is displayed in the table below. 

\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex} \ \rule[-1ex]{0pt}{0pt} & \textbf{Temperature (°C)} \\
\hline
\rule{0pt}{2.5ex} \text{Minimum} \rule[-1ex]{0pt}{0pt} & 16 \\
\hline
\rule{0pt}{2.5ex} Q_1 \rule[-1ex]{0pt}{0pt} & 21 \\
\hline
\rule{0pt}{2.5ex} \text{Median} \rule[-1ex]{0pt}{0pt} & 25 \\
\hline
\rule{0pt}{2.5ex} Q_3 \rule[-1ex]{0pt}{0pt} & 31 \\
\hline
\rule{0pt}{2.5ex} \text{Maximum} \rule[-1ex]{0pt}{0pt} & 39 \\
\hline
\end{array}

  1. Use the five-number summary above to construct a boxplot on the grid below.   (1 mark)
      


  1. Show, using calculations, that there are no outliers in the dataset.   (2 marks)

Show Answers Only
Show Worked Solution
a.   

 

b.   `IQR = Q_3-Q_1=31-21=10`

`text(Lower fence)` `= Q_1-1.5 xx IQR`
  `= 21-1.5 xx 10`
  `= 6`

 

`text(Upper fence)` `= Q_3 + 1.5 xx IQR`
  `= 31 + 1.5 xx 10`
  `= 46`

 

`text{Since 6 < 16 (minimum) and 46 > 39 (maximum)}`

`=>\ \text{There are no outliers.}`

Statistics, STD2 S1 2019 VCE-G 2*

The five-number summary below was determined from the sleep time, in hours, of a sample of 59 types of mammals.

\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \textbf{Statistic} \rule[-1ex]{0pt}{0pt} & \textbf{Sleep time (hours)} \\
\hline
\rule{0pt}{2.5ex} \text{minimum} \rule[-1ex]{0pt}{0pt} & \text{2.5} \\
\hline
\rule{0pt}{2.5ex} \text{first quartile} \rule[-1ex]{0pt}{0pt} & \text{8.0} \\
\hline
\rule{0pt}{2.5ex} \text{median} \rule[-1ex]{0pt}{0pt} & \text{10.5} \\
\hline
\rule{0pt}{2.5ex} \text{third quartile} \rule[-1ex]{0pt}{0pt} & \text{13.5} \\
\hline
\rule{0pt}{2.5ex} \text{maximum} \rule[-1ex]{0pt}{0pt} & \text{20.0} \\
\hline
\end{array}

  1. Show with calculations, that a boxplot constructed from this five-number summary will not include outliers.   (2 marks)

  2. Construct the boxplot below.   (1 mark)

     

Show Answers Only
  1. `text(Proof (See Worked Solution))`
  2.  
Show Worked Solution

a.    `IQR = Q_3-Q_1 = 13.5-8.0 = 5.5`

`text(Lower fence)` `= Q_1-1.5 xx IQR`
  `= 8-1.5 xx 5.5`
  `= -0.25`

 

`text(Upper fence)` `= Q_3 + 1.5 xx IQR`
  `= 13.5 + 1.5 xx 5.5`
  `= 21.75`

 
`text(S) text(ince) \ -0.25 < 2.5 \ text{(minimum value) and} \ 21.75 > 20.0 \ text{(maximum value)}`

`=> \ text(no outliers)`
 

b.