WORKSHEET General 2 MathematicsTopic Areas:Data and Statistics
DS2/3/4 – Single Data Sets, Summary Stats and Interpretation Stem & Leaf, Box & Whisker
Teacher: PETER HARGRAVESSource: HSC exam questionsExam Equivalent Time: 45 minutesWorked Solutions: IncludedNote: Each question has designated marks. Use this information as both a guide to the question's difficultyand as a timing indicator, whereby each mark should equate to 1.5 minutes of working (examination) time.
Questions
1. Data, 2UG 2012 HSC 1 MC
A set of scores is displayed in a stemandleaf plot.
What is the median of these scores?
(A)
(B)
(C)
(D)
15
7
8
77
78
2. Data, 2UG 2010 HSC 16 MC
This backtoback stemandleaf plot displays the test results for a class of students.
What is the median test result for the class?
(A)
(B)
(C)
(D)
3. Data, 2UG 2008 HSC 3 MC
The stemandleaf plot represents the daily sales of soft drink from a vending machine.
If the range of sales is , what is the value of ?
(A)
(B)
(C)
(D)
26
44
46
48
49
43
4
5
24
25
What is the mode for this data? (1 mark)
What is the median for this data? (1 mark)
4. Data, 2UG 2011 HSC 7 MC
A set of data is displayed in this boxandwhisker plot.
Which of the following best describes this set of data?
(A) Symmetrical
(B) Positively skewed
(C) Negatively skewed
(D) Normally distributed
5. Data, 2UG 2009 HSC 24a
The diagram below shows a stemandleaf plot for scores.
(i)
(ii)
22
Find the interquartile range for boys. (1 mark)
What percentage of girls usually spend or less hours on the internet over aweekend? (1 mark)Jenny said that the graph shows that the same number of boys as girls usuallyspend between and hours on the internet over a weekend.
Under what circumstances would this statement be true? (1 mark)
6. Data, 2UG 2009 HSC 26a
In a school, boys and girls were surveyed about the time they usually spend on the internetover a weekend. These results were displayed in boxandwhisker plots, as shown below.
(i)
(ii)
(iii)
5
5 6
What is the outlier for this set of data? (1 mark)
What is the interquartile range of the data collected from the female students? (1mark)
7. Data, 2UG 2011 HSC 25d
Data was collected from students on the number of text messages they had sent in theprevious 24 hours. The set of data collected is displayed.
(i)
(ii)
30
What is the modal travel time when he uses roads without tolls? (1 mark)
What is the median travel time when he uses roads without tolls? (1 mark)
Describe how the two data sets differ in terms of the spread and skewness of theirdistributions. (2 marks)
8. Data, 2UG 2013 HSC 26f
Jason travels to work by car on all five days of his working week, leaving home at eachday. He compares his travel times using roads without tolls and roads with tolls over a periodof working weeks.
He records his travel times (in minutes) in a backtoback stemandleaf plot.
(i)
(ii)
(iii)
7 am
12
Kim’s number summary for the tests is , , , , .
Draw a boxandwhisker plot to display Kim’s results below that of Terry’s results. (1 mark)
What percentage of Terry’s results were below ? (1 mark)
Terry claims that his results were better than Kim’s. Is he correct? Justify youranswer by referring to the summary statistics and the skewness of the distributions. (4 marks)
9. Data, 2UG 2014 HSC 29c
Terry and Kim each sat twenty class tests. Terry’s results on the tests are displayed in the boxandwhisker plot shown in part (i).
(i)
(ii)
(iii)
5 67 69 71 73 75
69
In 2000 there were children aged 0–18 years.
The number of children aged 12–18 years is the same in both 2000 and 2010.
How many children were aged 12–18 years in 2000? (1 mark)
How many children aged 0–18 years are there in 2010? (1 mark)
Identify TWO changes in the distribution of ages between 2000 and 2010. In youranswer, refer to measures of location or spread or the shape of the distributions. (2marks)
What would be ONE possible implication for government planning, as aconsequence of this change in the distribution of ages? (1 mark)
10. Data, 2UG 2010 HSC 27b
The graphs show the distribution of the ages of children in Numbertown in 2000 and 2010.
(i)
(ii)
(iii)
(iv)
1750
What is the interquartile range for English? (1 mark)
Compare and contrast the two data sets by referring to the skewness of thedistributions and the measures of location and spread. (3 marks)
11. Data, 2UG 2012 HSC 28d
The test results in English and Mathematics for a class were recorded and displayed in theboxandwhisker plots.
(i)
(ii)
Copyright © 200914 The State of New South Wales (Board of Studies, Teaching and Educational Standards NSW)
♦♦ Mean mark 35%
♦ Mean mark 47%
Worked Solutions
1. Data, 2UG 2012 HSC 1 MC
2. Data, 2UG 2010 HSC 16 MC
3. Data, 2UG 2008 HSC 3 MC
4. Data, 2UG 2011 HSC 7 MC
15 scores ⇒ Median is 8th
∴ Median is 78
⇒ D
26 results given in the data
⇒ Median is average of and 13th 14th
∴ Median =45 + 47
2= 46
⇒ B
Range = High−Low = 43
∴ 67 − Low = 43
Low = 24∴ N = 4
⇒ A
Since the median (155) is closer to the lower
quartile (150) and lower extreme (140) than the
upper equivalents, it is positively skewed.
⇒ B
♦♦ Mean mark 31%
♦♦♦ Mean mark 9%
♦♦ Mean mark 34%COMMENT: Ensure you canquickly and accurately findquartile values using stem andleaf graphs!
5. Data, 2UG 2009 HSC 24a
(i)
(ii)
6. Data, 2UG 2009 HSC 26a
(i)
(ii)
(iii)
7. Data, 2UG 2011 HSC 25d
(i)
(ii)
Mode = 78
22 scores
⇒ Median is the average of 11th and 12th scores
∴ Median =45 + 47
2= 46
Interquartile range = 6 −2
= 4
Upper quartile = 5
∴ 75% of girls spend 5 or less hours
5-6 hours for girls accounts for 25% of all girls.
5-6 hours for boys accounts for 25% of all boys,
(median to the upper quartile represents 25%.)
⇒ This will only be the same number if the number of
all girls surveyed equals the number of boys surveyed.
Outlier is 71
Lower quartile = 9 (4th female data point)
Upper quartile = 20 (11th female data point)
∴ Interquartile range (female) = 20 − 11 = 9
♦ Mean mark 36%MARKER'SCOMMENT: Finding amedian proved challenging formany students. Remember thatfor an even number of values,average the middle two asshown.
♦ Mean mark 39%
8. Data, 2UG 2013 HSC 26f
(i)
(ii)
(iii)
Modal time = 52 minutes
30 times with no tolls
Median = average and 15th 16th
=50 + 51
2= 50.5 minutes
Spread
Times without tolls have a much tighter
spread (range = 22) than times with tolls
(range = 55).
Skewness
Times without tolls shows virtually no skewness
while times with tolls are positively skewed.
♦ Mean mark 39%
♦♦ Mean mark 29%COMMENT: Examiners lookfavourably on using language oflocation in answers, particularlythe areas they have specificallypointed students towards(skewness in this example).
9. Data, 2UG 2014 HSC 29c
(i)
(ii)
(iii)
50 %
Terry's results are more positively skewed than
Kim's and also have a higher limit high.
However, Kim's results are more consistent,
showing a tighter IQR. They also have a
significantly higher median than Terry's and
are evenly skewed.
∴ Kim's results were better.
♦ Mean mark 45%
♦♦ Mean mark 25%
♦ Mean mark 35%MARKER'S COMMENT: Anumber of students incorrectlyidentified "positive" skew as"negative" skew here.
♦ Mean mark 46%MARKER'S COMMENT: This isworth only 1 mark and longwinded answers are notrequired.
10. Data, 2UG 2010 HSC 27b
(i)
(ii)
(iii)
(iv)
Since the median = 12 years
⇒ 50% of children are aged 12-18 years
∴ # Children 12-18 = 50 % × 1750
= 875
Upper quartile (2010) = 12 years
# Children in upper quartile = 875(from part (i))
∴ # Children aged 0-18 = 4 × 875
= 3500
Changes in distribution (only 2 req'd)
- the lower quartile age is lower in 2010
- the median is lower in 2010
- the upper quartile age is lower in 2010
- the interquartile range is greater in 2010
- 2010 is positively skewed while 2000 is negatively
Implication for government planning
- since the children are getting younger in 2010,
approve and build more childcare facilities
- Build more school and public playgrounds.
♦ Mean mark 35%MARKER'SCOMMENT: Markers arelooking for students to use thecorrect language of location andspread such as mean, median,interquartile range, standarddeviation and skewness.
11. Data, 2UG 2012 HSC 28d
(i)
(ii)
Copyright © 2015 M2 Mathematics Pty Ltd (SmarterMaths.com.au)
IQR (E) = 80 − 50
= 30
Skewness
- English has greater negative skew
- Maths is more normally distributed
Location and Spread
- English has a range of 85, Maths has 40.
- English has larger IQR than Maths (30 vs 15)
- Maths’ Median (75) is higher than English (70)
- Same upper quartile marks (80)
- English has highest and lowest individual mark