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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 053

The following ordered stem plot shows the percentage of homes connected to broadband internet for 24 countries in 2007.
 

   CORE, FUR1 2013 VCAA 1-2 MC 
 

  1. How many of these countries had more than 22% of homes connected to broadband internet in 2007?  (1 mark)
  2. What was the median percentage of homes connected to broadband?  (1 mark)
  3. For these countries, what is the modal percentage of homes connected to broadband?  (1 mark)
Show Answers Only

a.    \(19\)

b.    \(29.5\)

c.    \(31\)

Show Worked Solution

a.    \(\text{There are 19 values greater than 22%}\)

 

b.     \(\text{Median}\) \(=\dfrac{\text{12th + 13th}}{2}\)
    \(=\dfrac{29+30}{2}\)
    \(=29.5\)

 
c.    \(\text{Mode} = 31\)

Data Analysis, SM-Bank 052 MC

The stem plot below displays the average number of decayed teeth in 12-year-old children from `31` countries.

\begin{array} {r|lll}
\textbf{Stem} & \textbf{Leaf} \\
\hline 0 & 2 \\
0 & 5\ 6\ 7\ 7 \ 8\ 9\  \\
1 & 0\ 0\ 0\ 0\ 1\ 4\ 4\ 4\\
1 & 5\ 6\ 7 \\
2 & 3\ 3\ 4 \\
2 & 7\ 7\ 8\ 8 \\
3 & 0\ 4 \\
3 & 5\ 6 \\
4 & 1 \\
4 & 7 \\
\end{array}		 \(\text{key:  }0|2=0.2\)

Based on this stem plot, the distribution of the average number of decayed teeth for these countries is best described as

  1. positively skewed with a median of 15 decayed teeth and a range of 45
  2. approximately symmetric with a median of 1.5 decayed teeth and a range of 4.5
  3. negatively skewed with a median of 1.5 decayed teeth and a range of 4.5
  4. positively skewed with a median of 1.5 decayed teeth and a range of 4.5
Show Answers Only

\(D\)

Show Worked Solution

\(\text{Median = 16th value} = 1.5\)

\(\text{Range} = 4.7-0.2=4.5\)

\(\text{The clear tail to the upper end of values shows that the}\)

\(\text{data is positively skewed.}\)

\(\Rightarrow D\)

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 042

The following data represents the ages of people on a riverboat cruise in Europe.
 

28,  35,  39,  39,  28,  34,  40,  43,  51,  34,  35,  39,  40,  46,  60

 

  1. Organise the data into an ordered stem-and-leaf plot.  (2 marks)

  2. What is the median of the ages?  (1 mark)

  3. What is the modal age of the travellers?  (1 mark)

  4. Is this age data skewed or symmetrical?  (1 mark)

Show Answers Only

a.   

\begin{array} {r|lll} \textbf{Stem} & \textbf{Leaf}  \\ \hline  2 & 8\   8 \\  3 & 4\ 4\ 5\ 5 \ 9\ 9\ 9\  \\  4 & 0\ 0\ 3\ 6 \\ 5 & 1 \\ 6 & 0 \\ \end{array}

b.    `39`

c.    `39`

d.    `text(Skewed)`

Show Worked Solution

a.    `text(Ordered data:) \ 28,  28,  34,  34,  35,  35,  39,  39,  39,  40,  40,  43,  46,  51,  60`

\begin{array} {r|lll} \textbf{Stem} & \textbf{Leaf}  \\ \hline  2 & 8\   8 \\  3 & 4\ 4\ 5\ 5 \ 9\ 9\ 9\  \\  4 & 0\ 0\ 3\ 6 \\ 5 & 1 \\ 6 & 0 \\ \end{array}

b.    `text(Median)\ = \text(8th score)\ =\ 39`

c.    `text(Mode)\ = \ 39`

d.    `text(The data is skewed.)`

 

Data Analysis, SM-Bank 019

A vet measured the length of 21 dogs that came through his clinic.

The vet recorded the length of each dog.
  

  1.  What is the median length?  (1 mark)

  2. The vet measured and recorded a new dog with a length of 41 centimetres. What is the new median length?  (1 mark)

Show Answers Only

a.   `text(52 cm)`

b.   `text(51.5 cm)`

Show Worked Solution

a.   `text(The median of 21 data points is the 11th data point.)`

`:.\ text(The median is 52 cm.)`

 

b.   `text(The median of 22 data points is the average of the 11th and 12th data point.)`

`text(Average)` `=(51 +52)/2`
  `=51.5\ text(cm)`

 
`:.\ text(The new median is 51.5 cm.)`

Data Analysis, SM-Bank 015

A policeman is recording the speed of 25 cars travelling on a highway using a speed gun.

The results are shown in the stem-and-leaf plot.

  1. What is the median speed?  (1 mark)

  2. What is the range of the speeds?  (1 mark)

Show Answers Only

a.   `119\ text(km/hr)`

b.  `49\ text(km/hr)`

Show Worked Solution

a.   `25\ text(data points)`

`text(Median is the 13th data point)`

`:.\ text(Median) = 119\ text(km/hr)`
 

b.   `text(Range)` `= text(Highest) \ – \ text(Lowest)`
    `= 139\ – \ 90`
    `=49\ text(km/hr)`

 

Data Analysis, SM-Bank 014 MC

The Stars and the Thunder are playing cricket in a 20 over competition.

The stem-and-leaf plots show the number of runs each side has scored in their last 15 games.
 

   

Which statement is true about the data.

  1. The Stars had the lowest run score.
  2. The Stars scored over 135 runs more times than the Thunder.
  3. The median score for The Stars is higher than the median score for The Thunder.
  4. The range of scores for The Stars is smaller than the range of scores for The Thunder.
Show Answers Only

`D`

Show Worked Solution

`text{Range The Stars} = 149\ – \ 140\ =\ 39`

`text{Range The Thunder} = 148\ – \ 108\ =\ 40`

`:.\ text(Correct statement is:)`

`text{The range of scores for The Stars is smaller than the}`

`text{range of scores for The Thunder (39 vs 40).}`

`=>D`

 

Statistics, STD2 S1 2021 HSC 3 MC

The stem-and-leaf plot shows the number of goals scored by a team in each of ten netball games.
  

What is the mode of this dataset?

  1.  5
  2.  18
  3.  25
  4.  29
Show Answers Only

`C`

Show Worked Solution

`\text{Mode}  -> \text{data point with highest frequency}`

`\text{Mode}  = 25`

`=> C`

Statistics, STD2 S1 2006 HSC 4 MC

A set of scores is displayed in a stem-and-leaf plot.
 

 2UG-2006-4MC

 
What is the median of this set of scores?

  1.   28
  2.   30
  3.   33
  4.   47
Show Answers Only

`C`

Show Worked Solution

`text(10 scores)`

`text(Median)` `= text{5th + 6th}/2`
  `= (28 + 38)/2`
  `= 33`

`=>  C`

Statistics, STD2 S1 2008 HSC 3 MC

The stem-and-leaf plot represents the daily sales of soft drink from a vending machine.

If the range of sales is 43, what is the value of  2008 3 mc  ?

 
 

  1.    `4` 
  2.    `5`
  3.    `24`
  4.    `25`
Show Answers Only

`A`

Show Worked Solution

`text(Range = High) – text(Low) = 43`

`:.\ 67 – text(Low)` `= 43`
`text(Low)` `= 24`

`:.\ N = 4`

`=>  A`

Statistics, STD2 S1 2013 HSC 26f

Jason travels to work by car on all five days of his working week, leaving home at 7 am each day. He compares his travel times using roads without tolls and roads with tolls over a period of 12 working weeks.

He records his travel times (in minutes) in a back-to-back stem-and-leaf plot.
 

2013 26f
 

  1. What is the modal travel time when he uses roads without tolls?  (1 mark)

  2. What is the median travel time when he uses roads without tolls?   (1 mark)

  3. Describe how the two data sets differ in terms of the spread and skewness of their distributions.   (2 marks)

Show Answers Only
  1. `52\ text(minutes)`
  2. `50.5\ text(minutes)`
  3. `text(Spread)`
  4. `text{Times without tolls have a tighter spread (range = 22)}`
  5. `text{than times with tolls (range = 55).}`

  6.  

    `text(Skewness)`

  7. `text(Times without tolls shows virtually no skewness while`
  8. `text(times with tolls are positively skewed.)`
Show Worked Solution

i.  `text(Modal time) = 52\ text(minutes)`

♦ Mean mark 36%
MARKER’S COMMENT: Finding a median proved challenging for many students. Take note!

 

ii.  `text(30 times with no tolls)`

`text(Median)` `=\ text(Average of 15th and 16th)`
  `=(50 + 51)/2`
  `= 50.5\ text(minutes)`

 

♦ Mean mark 39%

 

 iii.  `text(Spread)`

`text{Times without tolls have a much tighter}`

`text{spread (range = 22) than times with tolls}`

`text{(range = 55).}`

`text(Skewness)`

`text(Times without tolls shows virtually no skewness)`

`text(while times with tolls are positively skewed.)`

Statistics, STD2 S1 2010 HSC 16 MC

This back-to-back stem-and-leaf plot displays the test results for a class of 26 students.
 

2010 Q16 MC  
 

What is the median test result for the class?

  1.    `44`
  2.    `46`
  3.    `48`
  4.    `49`
Show Answers Only

`B`

Show Worked Solution
♦♦ Mean mark 35%

`text(26 results given in the data)`

  `=>text(Median is average of)\ 13^text(th)\ text(and)\ 14^text(th)`

`:.\ text(Median)` `=(45+47)/2`
  `=46`

`=>B`

Statistics, STD2 S1 2009 HSC 24a

The diagram below shows a stem-and-leaf plot for 22 scores. 
 

2UG-2009-24a
 

  1.  What is the mode for this data?   (1 mark)

  2.  What is the median for this data?     (1 mark)

Show Answers Only
  1. `78`
  2. `46`
Show Worked Solution

i.   `text(Mode) = 78`

 

ii.    `22\ text(scores)`

`=>\ text(Median is the average of 11th and 12th scores)`
 

`:.\ text(Median)` `= (45 + 47)/2`
  `= 46`

Statistics, STD2 S1 2012 HSC 1 MC

A set of 15 scores is displayed in a stem-and-leaf plot.
 

2012 1 mc 
 

 What is the median of these scores?

  1.    7 
  2.    8
  3.   77
  4.   78
Show Answers Only

`D`

Show Worked Solution

`text(15 scores)\ \ =>\ \ text(Median is 8th)`

`:.\ \ text(Median is)\ \ 78`

`=>  D`