Displaying Data – Bar Charts and Histograms

Statistics, STD2 EQ-Bank 26

The image shows the proportion of people in various categories based on the 2021 census of the Australian population.

The number of people in the 2021 Australian census was $25\,418\,009$.

Of these, $812\,728$ identified as Aboriginal and/or Torres Strait Islander people.

Of those who identified as Aboriginal and/or Torres Strait Islander people, 51.1% were aged under 25 years and $243\,818$ were aged 10–24 years.

What percentage of the Gen Alpha category identified as Aboriginal and/or Torres Strait Islander people? (3 marks)

--- 7 WORK AREA LINES (style=lined) ---

Show Answers Only

$\text{Of those who identified as Aboriginal and/or Torres Strait Islander people }$

$\text{Number of people under 25 years }=812\,728 \times 0.511=415\,304$

$\text{Number in Gen Z}=243\,818 \ \text{(given)}$

$\text{Number in Gen Alpha}=415\,304-243\,818=171\,486$

$\text{Total number in Gen Alpha}=25\,418\,009 \times 0.12=3\,050\,161$

$\text{Percentage}=\dfrac{171\,486}{3\,050\,161}=5.6 \%$

Show Worked Solution

$\text{Of those who identified as Aboriginal and/or Torres Strait Islander people }$

$\text{Number of people under 25 years }=812\,728 \times 0.511=415\,304$

$\text{Number in Gen Z}=243\,818 \ \text{(given)}$

$\text{Number in Gen Alpha}=415\,304-243\,818=171\,486$

$\text{Total number in Gen Alpha}=25\,418\,009 \times 0.12=3\,050\,161$

$\text{Percentage}=\dfrac{171\,486}{3\,050\,161}=5.6 \%$

Statistics, STD2 S1 2025 HSC 28

The heights of students in a class were recorded.

The results for this class are displayed in the cumulative frequency graph shown.

The shortest student in this class is 130 cm and the tallest student is 180 cm.

Construct a box-plot for this class in the space below. (3 marks)

--- 7 WORK AREA LINES (style=lined) ---

Show Answers Only

Show Worked Solution

$Q_1(7.5 \ \text{students })=135$

$Q_3(22.5 \ \text{students })=160$

$\text{Median (15 students )}=140$

♦ Mean mark 50%.

Statistics, STD2 S1 2022 HSC 15 MC

The cumulative frequency graph shows the distribution of the number of movie downloads made by 100 people in one month.

Which box-plot best represents the same data as displayed in the cumulative frequency graph?

Show Answers Only

`C`

Show Worked Solution

`text{1st quartile}\ ~~ 3`

`text{Median}\ ~~ 6`

`text{3rd quartile}\ ~~ 7`

`=>C`

♦♦♦ Mean mark 30%.

Statistics, STD2 S1 2021 HSC 7 MC

The number of downloads of a song on each of twenty consecutive days is shown in the following graph.

Which of the following graphs best shows the cumulative number of downloads up to and including each day?

Show Answers Only

`C`

Show Worked Solution

`text{The gradient of the cumulative frequency histogram}`

♦ Mean mark 50%.

`text{will increase gradually, be steepest at day 10 then}`

`text{decrease gradually.}`

`=> C`

Statistics, STD2 S1 2020 HSC 7 MC

Which histogram best represents a dataset that is positively skewed?

Show Answers Only

`A`

Show Worked Solution

♦♦ Mean mark 32%.

`text(Positive skew occurs when the tail on the)`

`text{histogram is longer on the right-hand}`

`text{(positive) side.}`

`=> A`

Statistics, STD2 S1 2018 HSC 26d

The graph displays the mean monthly rainfall in Sydney and Perth.

For how many months is the mean monthly rainfall higher in Perth than in Sydney? (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
For which of the two cities is the standard deviation of the mean monthly rainfall smaller? Justify your answer WITHOUT calculations. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

a. `3`

b. `text(The mean monthly rainfall of Sydney is in a)`

`text(much tighter range than Perth.)`

`:.\ text(Sydney has a smaller standard deviation.)`

Show Worked Solution

a. `text(3 months (Jul, Aug and Sep))`

♦ Mean mark part (b) 38%.
COMMENT: Extremely volatile result between parts with part (a) producing a 91% mean mark.

b. `text(The mean monthly rainfall of Sydney is in a)`

`text(much tighter range than Perth.)`

`:.\ text(Sydney has a smaller standard deviation.)`

Statistics, STD2 S1 2016 HSC 27c

The heights of 400 students were measured. The results are displayed in this cumulative frequency polygon.

Use the polygon to estimate the interquartile range. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`36\ text(cm)`

Show Worked Solution

`text(See graph for values:)`

♦ Mean mark 46%.

`IQR= Q_3-Q_1= 172-136=36\ text(cm)`

Statistics, STD2 S1 2015 HSC 29d

Data from 200 recent house sales are grouped into class intervals and a cumulative frequency histogram is drawn.

Use the graph to estimate the median house price. (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
By completing the table, calculate the mean house price. (3 marks)

\begin{array} {|c|c|}
\hline
\rule{0pt}{2.5ex} \text{Class Centre} \rule[-1ex]{0pt}{0pt} & \text{Frequency} \\ \text{(\$'000)} & \\
\hline
\rule{0pt}{2.5ex} \rule[-1ex]{0pt}{0pt} & \\
\hline
\rule{0pt}{2.5ex} \rule[-1ex]{0pt}{0pt} & \\
\hline
\rule{0pt}{2.5ex} \rule[-1ex]{0pt}{0pt} & \\
\hline
\rule{0pt}{2.5ex} \rule[-1ex]{0pt}{0pt} & \\
\hline
\end{array}

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

a. `text(See Worked Solutions.)`

b. `text(See Worked Solutions.)`

Show Worked Solution

`text(From the graph, the estimated median)`

`text(house price = $392 500)`

♦♦♦ Mean marks of 9% for part (i) and 34% for part (ii)!

\begin{array} {|c|c|}
\hline
\rule{0pt}{2.5ex} \text{Class Centre} \rule[-1ex]{0pt}{0pt} & \text{Frequency} \\ \text{(\$’000)} & \\
\hline
\rule{0pt}{2.5ex} 375 \rule[-1ex]{0pt}{0pt} & 30\\
\hline
\rule{0pt}{2.5ex} 385 \rule[-1ex]{0pt}{0pt} & 50 \\
\hline
\rule{0pt}{2.5ex} 395 \rule[-1ex]{0pt}{0pt} & 70 \\
\hline
\rule{0pt}{2.5ex} 405 \rule[-1ex]{0pt}{0pt} & 50 \\
\hline
\end{array}

`text(Mean house price ($’000))`

`= (375xx30 + 385xx50 + 395xx70 + 405xx50)/200`

`=$392`

`:. text(Mean house price is)\ $392\ 000`

Statistics, STD2 S1 2006 HSC 24c

The heights of the 60 members of a choir were recorded. These results were grouped and then displayed as a cumulative frequency histogram and polygon.

The shortest person in the choir is 140 cm and the tallest is 190 cm.

Draw an accurate box-and-whisker plot to represent the data. (3 marks)

--- 8 WORK AREA LINES (style=lined) ---

Show Answers Only

`text{See Worked Solutions}`

Show Worked Solution

`text(Low) = 140`

`text(High) = 190`

`text(Median) = 150\ \ \ \ text{(# People = 30)}`

`Q_1 = 145\ \ \ \ text{(# People = 15)}`

`Q_3 = 170\ \ \ \ text{(# People = 45)}`

`text(Box and Whisker)`

Statistics, STD2 S1 2006 HSC 24a

List TWO ways in which this graph is misleading. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(Reasons the graph is misleading include)`

`text(- the columns are of different widths/volumes)`

`text(- the vertical axis doesn’t start at zero)`

Show Worked Solution

`text(Reasons the graph is misleading include)`

`text(- the columns are of different widths/volumes)`

`text(- the vertical axis doesn’t start at zero)`

Statistics, STD2 S1 2005 HSC 24d

The sector graph shows the proportion of people, as a percentage, living in each region of Sumcity. There are 24 000 people living in the Eastern Suburbs.

Show that the total number of people living in Sumcity is 160 000. (1 mark)

--- 1 WORK AREA LINES (style=lined) ---

Jake used the information above to draw a column graph.

The column graph height is incorrect for one region.

Identify this region and justify your answer. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

a. `160\ 000`

b. `text(Western Suburbs population)`

Show Worked Solution

a. `text(Let the population of Sumcity =)\ P`

`text(15%)× P`	`= 24\ 000`
`:.P`	`= (24\ 000)/0.15= 160\ 000\ …\ text(as required)`

b. `text(Western Suburbs population)`

`= text(10%) × 160\ 000= 16\ 000`

`text(The column graph has this population as)`

`text(12 000 people which is incorrect.)`

Statistics, STD2 S1 2005 HSC 9 MC

A set of data is represented by the cumulative frequency histogram and ogive.

What is the best approximation for the interquartile range for this set of data?

Show Answers Only

`C`

Show Worked Solution

`IQR= Q3-Q1= 80-45= 35`

`=> C`

Statistics, STD2 S1 2008 HSC 26d

The graph shows the predicted population age distribution in Australia in 2008.

How many females are in the 0–4 age group? (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
What is the modal age group? (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
How many people are in the 15–19 age group? (2 marks)

--- 4 WORK AREA LINES (style=lined) ---
Give ONE reason why there are more people in the 80+ age group than in the 75–79 age group. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

a. `600\ 000`

b. `35-39`

c. `1\ 450\ 000`

d. `text(The 80+ group includes all people over 80)`

`text(and is not restricted by a 5-year limit.)`

Show Worked Solution

a. `text{# Females} (0-4)= 0.6 xx 1\ 000\ 000= 600\ 000`

b. `text(Modal age group)\ =35-39`

c. `text{# Males} (15-19)= 0.75 xx 1\ 000\ 000= 750\ 000`

`text{# Females} (15-19)= 0.7 xx 1\ 000\ 000= 700\ 000`

`:.\ text{Total People} (15-19)`	`= 750\ 000 + 700\ 000`
	`= 1\ 450\ 000`

d.	`text(The 80+ group includes all people over 80)`
	`text(and is not restricted by a 5-year limit.)`

Statistics, STD2 S1 2014 HSC 26e

The times taken for 160 music downloads were recorded, grouped into classes and then displayed using the cumulative frequency histogram shown.

On the diagram, draw the lines that are needed to find the median download time. (2 marks)

--- 0 WORK AREA LINES (style=lined) ---

Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 17%.

Statistics, STD2 S1 2011 HSC 25b

The graph below displays data collected at a school on the number of students
in each Year group, who own a mobile phone.

Which Year group has the highest percentage of students with mobile phones? (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
Two students are chosen at random, one from Year 9 and one from Year 10.

Which student is more likely to own a mobile phone?

Justify your answer with suitable calculations. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---
Identify a trend in the data shown in the graph. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

a. `text{Year 12 (100%)}`

b. `text(Year 10)`

c. `text(See Worked Solutions for detail)`

Show Worked Solution

a. `text(Year 12 (100%))`

MARKER’S COMMENT: Many students ignore the instruction to use calculations and lose marks as a result.

b.	`text(% Ownership in Year 9)`	`=55/70`
		`=\ text{78.6% (1d.p.)}`
	`text(% Ownership in Year 10)`	`=50/60`
		`=\ text{83.3% (1d.p.)}`

`:.\ text(The Year 10 student is more likely to own a mobile phone.)`

c. `text(% Ownership increases as students)`

`text(progress from Year 7 to Year 12.)`

Statistics, STD2 S1 2010 HSC 26b

A new shopping centre has opened near a primary school. A survey is conducted to determine the number of motor vehicles that pass the school each afternoon between 2.30 pm and 4.00 pm.

The results for 60 days have been recorded in the table and are displayed in the cumulative frequency histogram.

Find the value of Χ in the table. (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
On the cumulative frequency histogram above, draw a cumulative frequency polygon (ogive) for this data. (1 mark)
Use your graph to determine the median. Show, by drawing lines on your graph, how you arrived at your answer. (1 mark)
Prior to the opening of the new shopping centre, the median number of motor vehicles passing the school between 2.30 pm and 4.00 pm was 57 vehicles per day.

What problem could arise from the change in the median number of motor vehicles passing the school before and after the opening of the new shopping centre?

Briefly recommend a solution to this problem. (2 marks)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

a. `15`

d. `text(Problems)`

`text(- increased traffic delays)`

`text(- increased danger to students leaving school)`

`text(Solutions)`

`text(- signpost alternative routes around school)`

`text(- decrease the speed limit in the area)`

Show Worked Solution

a. `X= 25-10= 15`

♦♦♦ Mean mark 18%
MARKER’S COMMENT: The ogive was poorly drawn with many students incorrectly joining the middle of each column rather than from corner to corner.

♦♦ Mean mark 25%
MARKER’S COMMENT: Many students did not “show by drawing lines on the graph” as the question asked.

c. `text(Median)\ ~~155`

♦ Mean mark 47%
MARKER’S COMMENT: Short answers were often the best. Be concise when you can.

d.	`text(Problems)`
	`text(- increased traffic delays)`
	`text(- increased danger to students leaving school)`

	`text(Solutions)`
	`text(- signpost alternative routes around school)`
	`text(- decrease the speed limit in the area)`

Statistics, STD2 S1 2013 HSC 27c

A retailer has collected data on the number of televisions that he sold each week in 2012.

He grouped the data into classes and displayed the data using a cumulative frequency histogram and polygon (ogive).

Use the cumulative frequency polygon to determine the interquartile range. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---
Oscar said that the retailer sold 300 televisions in 6 of the weeks in 2012.

Is he correct? Give a reason for your answer. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

a. `280`

b. `text(Oscar is not correct because the data is grouped.)`

`text(He could only say that the retailer sold between 250`

`text(and 350 units in 6 of the weeks in 2012.)`

Show Worked Solution

a. `text(S)text(ince cumulative frequency = 52 at max)`

♦♦♦ Mean mark 13%
COMMENT: Finding median and quartile values from cumulative frequency charts is an important skill that is often examined.

`=> text(Lower quartile at week) = 52/4 = 13`

`text(Lower quartile = 190)\ \ \ \ text{(from graph)}`

`=> text(Upper quartile at week) = 3 xx 13 = 39`

`text(Upper quartile = 470)`

`:.\ text(IQR)= 470-190=280`

b. `text(Oscar is not correct because the data is grouped.)`

♦♦♦ This question proved a beast, with a mean mark of 1%. Toughest question in the 2013 exam!

`text(He could only say the retailer sold between)`

`text(250 and 350 units in 6 of the weeks in 2012.)`

Statistics, STD2 S1 2011 HSC 11 MC

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

Which of these statements is true?

`X` has a larger mode and `Y ` has a larger range.
`X` has a larger mode and the ranges are the same.
The modes are the same and `Y` has a larger range.
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`

Statistics, STD2 S5 2010 HSC 4 MC

Which of the following frequency histograms shows data that could be normally distributed?

Show Answers Only

`A`

Show Worked Solution

`text(Normally distributed data have a frequency)`

`text(histogram graph that is shaped like a bell.)`

`=> A`

Statistics, STD2 S1 2013 HSC 15 MC

The frequency histogram shows the number of goals scored by a football team in each game in a season.

What is the mean number of goals scored per game by this team?

Show Answers Only

`C`

Show Worked Solution

`text(Total number of goals scored)`

`=(3xx3)+(4xx7)+(5xx5)+(6xx1)+(7xx0)+(8xx4)`

`=9+28+25+6+0+32=100`

`text(Number of games)=3+7+5+1+4=20`

`:.\ text(Mean goals per game)=100/20=5`

`=>\ C`

Statistics, STD2 S1 2013 HSC 6 MC

A survey was conducted where people were asked which of two brands of smartphones they preferred. The results were:

48% preferred Brand X
52% preferred Brand Y

A graph displaying the data is to be included in a magazine article. The editor of the magazine wishes to ensure that the graph is not misleading in any way.

Which graph should the editor choose to include in the article?

Show Answers Only

`D`

Show Worked Solution

`D\ text(is the best as it starts at zero on the)\ y text(-axis)`

`text(and has the same column widths.)`

`=> D`