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Data Analysis, SM-Bank 055

The stem plot below shows the distribution of mathematics test scores for a class of 23 students.
 


 

For this class:

  1. What was the range of test scores?  (1 mark)
  2. What was the mean test score, correct to 1 decimal place?  (2 marks)
  3. What was the median test mark?  (1 mark)
  4. What was the mode of the test scores?  (1 mark)
  5. A student sits the test late and scores a mark of 58. Describe the change, if any, in the range, the mean, the median and the mode.  (2 marks)
Show Answers Only

a.    \(49\)

b.    \(64.3\ \text{(1 d.p.)}\)

c.    \(68\)

d.    \(\text{Range → unchanged}\)

\(\text{Mean → reduced}\)

\(\text{Median → reduced}\)

\(\text{Mode → unchanged}\)

Show Worked Solution
a.    \(\text{Range}\) \(=89-40\)
    \(=49\)

 

b.   \(\text{Mean}\) \(=\dfrac{40+41+2\times 44+52+57+3\times 59+65+66+2\times 68+2\times 69+2\times 70+75+76+77+78+85+89}{23}\)
    \(=\dfrac{1480}{23}\)
    \(=64.347\dots\)
    \(\approx 64.3\ \text{(1 d.p.)}\)

 

c.    \(\text{Median}\) \(=\dfrac{23+1}{2}\ \text{score}\)
    \(=\text{12th score}\)
    \(=68\)

 
d.    \(\text{Range}\ \longrightarrow\ \text{stays the same}\)
 

\(\text{Mean}\) \(=\dfrac{1480+58}{24}\)
  \(=64.1\ \text{(1 d.p.)}\)
  \(\therefore\ \text{Mean is reduced}\)

 

\(\text{Median}\) \(=\dfrac{\text{12th score+13th score}}{2}\)
  \(=\dfrac{66+68}{2}\)
  \(=67\)
  \(\therefore\ \text{Median is reduced}\)

 

\(\text{Mode}\ \longrightarrow\ \text{stays the same}\)

 

Data Analysis, SM-Bank 054

The following ordered stem plot shows the areas, in square kilometres, of 27 suburbs of a large city.

\begin{array} {r|lll}
\textbf{Stem} & \textbf{Leaf} \\
\hline 1 & 5\ 6\ 7\ 8  \\
2 & 1\ 2\ 4\ 5 \ 6\ 8\ 9\ 9 \\
3 & 0\ 1\ 1\ 2\ 2\ 8\ 9 \\
4 & 0\ 4\ 7 \\
5 & 0\ 1 \\
6 & 1\ 9 \\
7 &  \\
8 & 4 \\
\end{array}		 \(\text{key:  }1|6=1.6\ \text{km}^2\)
  1. For these suburbs
    i.     What is the median, in square kilometres?  (1 mark)

    ii.    What is the range, in square kilometres?  (1 mark)

  2. What is the possible outlier?  (1 mark)
  3. Briefly describe the skewness of the data.  (1 mark)
Show Answers Only

a.    i.    \(3.1\ \text{km}^2\)

ii.   \(6.9\)

b.    \(8.4\ \text{km}^2\)

c.    \(\text{Positively skewed.}\)

Show Worked Solution
a.    i.     \(\text{Median}\) \(=\dfrac{27+1}{2}\)
      \(=\ \text{14 th score}\)
    \(\therefore\ \text{Median}\) \(=3.1\ \text{km}^2\)
       
  ii. \(\text{Range}\) \(=8.4-1.5\)
      \(=6.9\)

 
b.    \(8.4\ \text{km}^2\ \text{is a possible outlier}\)

c.    \(\text{The data is positively skewed as the tail is to the right.}\)

Data Analysis, SM-Bank 051

Ms Granger measured and recorded the heights of all the students in her class, to the nearest centimetre.

She made a dot plot to show the heights of these 32 children.
 

   
Student Heights (nearest cm)

 

  1. What fraction of the students' heights are greater than 145 centimetres and less than 150?  (2 marks)
  2. What is the range of the heights?  (1 mark)
  3. What is the median of the heights?  (1 mark)
  4. What would the median be if a new student arrived with a height of 135 cm?  (1 mark)
Show Answers Only

a.    \(\dfrac{1}{4}\)

b.    \(18\)

c.    \(147.5\ \text{cm}\)

d.    \(147\ \text{cm}\)

Show Worked Solution
a.    \(\text{Fraction}\) \(=\dfrac{\text{Students with height 145 – 150}}{\text{Total students}}\)
    \(=\dfrac{8}{32}\)
    \(=\dfrac{1}{4}\)

 
\(\text{(Note students with heights of 145 or 150 are not included)}\)
 

b.    \(\text{Range}=154-136=18\)
 

c.    \(\text{Median}\) \(=\dfrac{\text{16th score + 17th score}}{\text{2}}\)
    \(=\dfrac{147+148}{2}\)
    \(=147.5\ \text{cm}\)

 

d.    \(\text{Median }\) \(=\text{ 17th score}\)
    \(=147\ \text{cm}\)

Data Analysis, SM-Bank 046

The back-to-back ordered stem-and-leaf plot below shows the female and male smoking rates, expressed as a percentage, in 18 countries.
 

  1. For the 18 countries listed, what is the range of the male smoking rates?  (1 mark)
  2. For the 18 countries listed, what is the mode of the female smoking rates?  (1 mark)
  3. For the 18 countries listed, what is the difference between the medians of the female and male smoking rates?  (2 marks)
Show Answers Only

a.     \(30\%\)

b.     \(25\%\)

c.     \(5.5\%\)

Show Worked Solution

a.    \(\text{Range}=47-17=30\%\)

b.    \(\text{Female mode}=25\%\)

c.     \(\text{Female Median }\) \(=\ \text{average of 9th and 10th scores}\)
    \(=\dfrac{21+22}{2}=21.5\%\)

 

\(\text{Male Median }\) \(=\ \text{average of 9th and 10th scores}\)
  \(=27\%\)

  
\(\therefore\ \text{The difference in medians}\)

\(=27-21.5=5.5\%\)

Data Analysis, SM-Bank 025

A shop sells children's shoes in sizes from 5 to 9. The sizes of the last 100 shoes sold is shown in the table below.

\begin{array} {|l|c|c|c|c|c|c|}
\hline \textbf{Shoe Size} & 5 & 6 & 7 & 8 & 9  \\
\hline \textbf{Frequency}  & 14 & 21 & 30 & 23 & 12 \\
\hline \end{array}

  1. What is the median shoe size?  (1 mark)

  2. What is the mean shoe size?  Give your answer correct to the nearest whole number.  (2 marks)

  3. What is the range of shoe sizes?  (1 mark)

  4. What is the modal shoe size?  (1 mark)

  5. Briefly explain why the shop owner would be particularly interested in the modal shoe size?  (2 marks)

Show Answers Only
  1. `7`
  2. `6.98`
  3. `4`
  4. `7`
  5. `text(The shop keeper would be primarily interested in profit,)`
    `text(and the most popular shoe size will generate the most profit.)`
    `text(Also for stock purposes it would be an advantage to have the)`
    `text(most popular size in stock.)`
Show Worked Solution

a.    `text(Median) \ = \text(Average of 50th and 51st scores) \ = 7`

b.    `text(Mean)` `=(5 xx 14 + 6 xx 21 + 7 xx 30 + 8 xx 23 + 9 xx 12)/100`  
  `=(70 + 126 + 210 + 184 +108)/100`  
  `=698/100 = 6.98`  

 
c.    `text(Range) \ = \text(highest score) \-\ text(lowest score) \ = 9 \-\ 5 = 4`

d.    `text(Modal shoe size) \ = \text(The shoe size with the highest frequency) \ = 7`

\begin{array} {ll}
\textbf{e.}&\text{The shop keeper would be primarily interested in profit,}\ \\
&\text{and the most popular shoe size will generate the most profit.}\ \\
&\text{Also for stock purposes it would be an advantage to have the}\ \\ 
&\text{most popular size in stock.}\end{array}

Data Analysis, SM-Bank 021 MC

A survey asked 9 students in Year 8 how many siblings they had.
 

\begin{array} {|c|c|}\hline \textbf{Number of Siblings}  & 0,  2,  2,  2,  3,  4,  4,  5,  5\\ \hline \end{array}

 
Which of the following is true for this data?

  1. mean > median > mode
  2. mean > median < mode
  3. mean = median = mode
  4. mean = median > mode
Show Answers Only

`D`

Show Worked Solution

`text(Mode = 2)`

`text(Median = 5th number = 3)`

`text(Mean)` `= (0 + 2 + 2 + 2 + 3 + 4 + 4 + 5 + 5)/9`
  `= 27 ÷ 9`
  `= 3`

 

`:.\ text(mean = median > mode)`

`=>D`

Data Analysis, SM-Bank 010 MC

The heights, in centimetres, of David's hockey side are displayed in the dot plot below.
 

Which of the following statements is true about this data?

  1. The median and the mode are both 174 and the mean is 174.5.
  2. The mean and the median are both 174 and the mode is 177.
  3. The mean is 175, the mode is 174 and the median is 174.5.
  4. The mean, median and mode are all equal to 174.
Show Answers Only

`D`

Show Worked Solution

`text(Data from the dot plot:)` `\ 171, \ 172, \ 172, \ 173, \ 174, \ 174, \ 174, \ 174, \ 176, \ 177, \ 177`

`text(Median)\ ` `=\ text(Middle or 6th score)`
  `= 174`

 

`text(Mode)\ ` `=\ text(The most frequent score)`
  `= 174`

 

`text(Mean)\ ` `=\ text(Average of the scores)`
  `= (171 + 172 + 172 + 173 + 174 + 174 + 174 + 174 + 176 + 177 + 177)/11`
  `= 1914/11`
  `= 174`

 
∴ The mean, median and mode are all equal to 174.

`=>D`

Data Analysis, SM-Bank 005 MC

The points scored by an AFL team in their first 13 games of the season is recorded.

`78, \ 84, \ 63, \ 75, \ 98, \ 105, \ 92, \ 75, \ 84, \ 96, \ 84, \102, \100`

In the 14th game, they scored 61.

Which of these values would increase?

  1. `text(mode)`
  2. `text(range)`
  3. `text(mean)`
  4. `text(median)`
Show Answers Only

`B`

Show Worked Solution

`text(Consider each option:)`

`text(Mode – unchanged at 84)`

`text(Range – increases from 42 to 44)`

`text(Mean – decreases from 87.38 to 85.5)`

`text{Median – unchanged at 84}`

`=>B`

Statistics, STD2 S1 2004 HSC 6-7 MC

Use the set of scores  1, 3, 3, 3, 4, 5, 7, 7, 12  to answer Questions 6 and 7.
 

Question 6

What is the range of the set of scores?

  1. 6
  2. 9
  3. 11
  4. 12

 

Question 7

What are the median and the mode of the set of scores?

  1. Median 3, mode 5
  2. Median 3, mode 3
  3. Median 4, mode 5
  4. Median 4, mode 3
Show Answers Only

`text(Question 6:)\ C`

`text(Question 7:)\ D`

Show Worked Solution

`text(Question 6)`

`text(Range)` `= text(High) – text(Low)`
  `= 12 – 1`
  `= 11`

`=> C`

 

`text(Question 7)`

`text(9 scores)`

`:.\ text(Median)` `= (9 + 1) / 2`
  `=5 text(th score)`
  `= 4`

`text(Mode) = 3`

`=> D`

Statistics, STD2 S1 2009 HSC 3 MC

The eye colours of a sample of children were recorded.

When analysing this data, which of the following could be found?

  1. Mean
  2. Median
  3. Mode
  4. Range
Show Answers Only

`C`

Show Worked Solution

`text(Eye colour is categorical data)`

`:.\ text(Only the mode can be found)`

`=>  C`

Statistics, STD2 S1 2011 HSC 11 MC

The sets of data, `X` and `Y`, are displayed in the histograms.

2UG 2011 11

Which of these statements is true?

  1.   `X` has a larger mode and `Y ` has a larger range.
  2.   `X` has a larger mode and the ranges are the same.
  3.   The modes are the same and `Y` has a larger range.
  4.   The modes are the same and the ranges are the same.
Show Answers Only

`B`

Show Worked Solution
♦ Mean mark 47%

`text(Mode of)\ X=9`

`text(Range of)\ X=9-3=6`

`text(Mode of)\ Y=8`

`text(Range of)\ Y=11-5=6`

`:. X\ text(has a larger mode and ranges are the same)`

`=>B`