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1.02 Describing data

Worksheet
Measures of centre
1

Find the mode of the following scores:

2, 2, 6, 7, 7, 7, 7, 11, 11, 11, 13, 13, 16, 16

2

A rating system of 1 - 3 was used in a survey to determine the usefulness of a new feature. The 10 scores shown below are known to have a mode of 1.

3, 2, 3, 2, 1, 3, 1, 1, 2, x

Find the missing score, x.

3

Find the median of 7, 4, 6, 3.

4

A set of 69 scores is arranged in ascending order. In what position does the median score lie?

5

In a set of 152 scores, between which two scores does the median lie?

6

Find the mean of the following sets of scores:

a

22.4, 25.4, 19.1, 24.3, 7.4

b

- 14, 0, - 2, - 18, - 8, 0, - 15, - 1.

7

The following five numbers have a mean of 11:

11, 13, 9, 13, 9

If a new number is added that is smaller than 9, describe the effect on the mean.

8

Durations of calls (in minutes) made in a household were recorded as follows:

5,\text{ }\text{ } 11,\text{ }\text{ } 3,\text{ }\text{ } 9,\text{ }\text{ } 5,\text{ }\text{ } 14,\text{ }\text{ } 5,\text{ }\text{ } 14,\text{ }\text{ } 14,\text{ }\text{ } 3,\text{ }\text{ } 7,\text{ }\text{ } 7,\text{ }\text{ } 7,\text{ }\text{ } 5,\text{ }\text{ } 3,\text{ }\text{ } 3,\text{ }\text{ } 7,\text{ }\text{ } 14,\text{ }\text{ } 5

a

What was the total number of calls made?

b

What was the longest duration of a call?

c

What was the shortest duration of a call?

d

What was the mean duration of a call, correct to two decimal places?

e

What was the modal duration?

f

What was the median duration?

9

A real estate agent wanted to determine a typical house price in a certain area. He gathered the selling price of some houses (in dollars):

317\,000, \text{ }\text{ }320\,000,\text{ }\text{ } 347\,000,\text{ }\text{ } 360\,000,\text{ }\text{ } 378\,000,\text{ }\text{ } 395\,000,\text{ }\text{ } 438\,000,\text{ }\text{ } 461\,000,\text{ }\text{ } 479\,000,\text{ }\text{ } 499\,000

a

Calculate the mean house price.

b

What percentage of the house prices exceed the mean?

c

Determine the median house price.

d

What percentage of house prices exceed the median?

10

Susanah has been growing watermelons. The weights of the watermelons (in kilograms) are: 15,\text{ }\text{ } 6,\text{ }\text{ } 5,\text{ }\text{ } 2,\text{ }\text{ } 4,\text{ }\text{ } 4,\text{ }\text{ } 5

a

Calculate the median weight of the watermelons.

b

Calculate the mean weight, correct to two decimal places.

c

Which measure of centre is a more accurate description of the centre of this data set? Explain your answer.

11

The median house price in Humbleton is \$950\,000 with a mean price of \$1\,000\,000, and the median house price in Brockway is \$950\,000 with a mean price of \$880\,000.

Which town overall has the more expensive houses? Explain your answer.

12

The stem and leaf plot shows the prices, in dollars, of concert tickets locally and internationally:

a

Find the most expensive ticket price at the international venue.

b

Find the median ticket price at the international venue, correct to two decimal places.

c

Find the percentage of local ticket prices that were cheaper than the international median.

d

At the international venue, calculate the percentage of tickets costing between \$90 and \$110.

International PricesLocal Prices
8 261 3 7 9
8 8 3 173 4 4 7 9
7 5 2 181 2 3 5 8
7 6 5 1 191 3 4 5 8
8 5 4 1 0104

Key: 2 \vert 6 \vert 0 = 62 \text{ and }60

e

At the local venue, calculate the percentage of tickets costing between \$90 and \$100.

Measures of spread
13

Find the range of the following sets of scores:

a

10, 7, 2, 14, 13, 15, 11, 4

b

15, - 2 , - 8 , 8, 15, 6, - 16 , 15

14

A group of students had a range in marks of 14 and the lowest score was 9. Determine the highest score in the group.

15

Consider the following set of scores:

10,\text{ } 11,\text{ } 12,\text{ } 13,\text{ } 15,\text{ } 17, \text{ }19,\text{ } 20

a

Within what range do the middle 50\% of scores lie?

b

State another name for the middle 50\% of scores.

16

Use the statistics mode on the calculator to determine the standard deviation of the following sets of scores. Round your answer to two decimal places.

a

- 17,\text{ } 2,\text{ } - 6 ,\text{ } 9,\text{ } - 17,\text{ } - 9,\text{ } 3,\text{ } 8,\text{ } 5

b

8, \text{ }20, \text{ }16, \text{ }9, \text{ }9, \text{ }15, \text{ }5, \text{ }17, \text{ }19, \text{ }6

17

Meteorologists predicted a huge variation in temperatures throughout the month of April. The temperature each day for the first two weeks of April were recorded as follows:

16,\text{ } 18,\text{ } 20.5,\text{ } 21,\text{ } 21,\text{ } 21, \text{ }21.5, \text{ }22, \text{ }22,\text{ } 24,\text{ } 24,\text{ } 25,\text{ } 26,\text{ } 27

a

State the range of the temperatures.

b

Calculate the interquartile range of the temperatures.

c

Use your calculator to determine the standard deviation. Round your answer to one decimal place.

d

Would the standard deviation or the interquartile range be the best measure of spread to support the prediction of a huge variation in temperatures? Explain your answer.

18

Consider the frequency distribution table below:

a

Complete the table.

b

Calculate the mean, correct to two decimal places.

c

State the mode.

d

Find the range.

e

Determine the number of scores that are less than the mode.

\text{Score } (x)\text{Frequency } (f)fx
411
535
16
14
\text{Total}43365
19

Consider the following set of scores and calculate:

13, \text{ }15,\text{ } 5, \text{ }16,\text{ } 7,\text{ } 20,\text{ } 12

a

The median.

b

The range.

c

The first quartile.

d

The third quartile.

e

The interquartile range.

20

Consider the following set of scores:

- 3,\text{ } - 3,\text{ } 1,\text{ } 9,\text{ } 9,\text{ } 6,\text{ } - 9

a

Find the median.

b

Find the first quartile.

c

Find the third quartile.

d

Calculate the interquartile range.

21

In competition, a diver must complete 8 rounds of dives. Her scores for the first 7 rounds are given below:

7.3,\text{ } 7.4,\text{ } 7.7,\text{ } 8.4,\text{ } 8.7,\text{ } 8.9,\text{ } 9.4

Determine her score in the 8th round if the upper quartile of all 8 scores is 8.85.

22

There is a test to measure the Emotional Quotient (EQ) of an individual. Below are the EQ results for 21 people, listed in ascending order:

92,\text{ } 94,\text{ } 100,\text{ } 103,\text{ } 103,\text{ } 105,\text{ } 105,\text{ } 109,\text{ } 110,\text{ } 113, \text{ } 114,

114,\text{ } 116,\text{ } 118,\text{ } 118,\text{ } 119,\text{ } 120,\text{ } 125,\text{ } 125,\text{ } 126,\text{ } 130

a

Find the median.

b

Find Q_1.

c

Find Q_3.

d

Calculate the interquartile range.

23

Consider the dot plot:

a

Determine the first quartile.

b

Determine the third quartile.

c

Calculate the interquartile range.

d

Find the range.

Compare two sets of data
24

10 participants had their pulse measured before and after exercise with results shown in the following stem and leaf plot:

a

Calculate the modal pulse rate after exercise.

b

How many modes are there for the pulse rate before exercise?

c

Calculate the range of pulse rates before exercise.

d

Calculate the range of pulse rates after exercise.

e

Calculate the mean pulse rate before exercise.

f

Calculate the mean pulse rate after exercise.

g

Explain the effect of exercise on pulse rates.

Pulse rate before exercisePulse rate after exercise
0 5 55
4 7 9 96
3 47
084
95 7 8
103
113 5 5
120 1

Key: 2 \vert 6 \vert 0 = 62 \text{ and }60

25

The beaks of two groups of birds are measured, in millimetres, to determine whether they might be of the same species:

Group 140413249343443473738
Group 255544444545443474139
a

Calculate the range for Group 1.

b

Calculate the range for Group 2.

c

Calculate the mean for Group 1, correct to one decimal place.

d

Calculate the mean for Group 2, correct to one decimal place.

e

Explain why the two groups of birds are most likely different species.

26

Marge grows two different types of bean plants. She records the number of beans that she picks from each plant for 10 days. Her records are as follows:

  • Plant A: 4,\text{ } 4, \text{ }5, \text{ }7, \text{ }10,\text{ } 3,\text{ } 3,\text{ } 9,\text{ } 10, \text{ } 10

  • Plant B: 8,\text{ } 7,\text{ } 5,\text{ } 5,\text{ } 9,\text{ } 7,\text{ } 8,\text{ } 7,\text{ } 5,\text{ } 6

a

Find the mean number of beans picked per day for Plant A, correct to one decimal place.

b

Find the mean number of beans picked per day for Plant B, correct to one decimal place.

c

Find the range for Plant A.

d

Find the range for Plant B.

e

Which plant produces more beans on average? Explain your answer.

f

Which plant has a more consistent yield of beans? Explain your answer.

27

Two English classes, each with 15 students, sit a ten question multiple choice test. Their class results, out of 10, are below:

Class 1333151242423321
Class 287108869879981089
a

Calculate the mean, median, mode and range for Class 1. Round your answers to one decimal place if necessary.

b

Calculate the mean, median, mode and range for Class 2. Round your answers to one decimal place if necessary.

c

Which class was more likely to have studied effectively for their test?

d

Explain how the statistical calculations support your answer.

28

The mean income of people in Finland is \$45\,000. This is the same as the mean income of people in Canada. The standard deviation of Finland is greater than the standard deviation of Canada. In which country is there likely to be the greatest difference between the incomes of the rich and poor? Explain your answer.

29

The table shows the number of goals scored by a football team in each game of the year:

a

In how many games were 0 goals scored?

b

Determine the median number of goals scored, correct to one decimal place.

c

Calculate the mean number of goals scored each game, correct to two decimal places.

d

Find the standard deviation, correct to two decimal places.

\text{Score }(x)\text{Frequency } (f)
03
11
25
31
45
55
30

Consider the histogram below:

a

Find the range of the data set.

b

Find the mean of the data set. Round your answer to two decimal places.

c

Find the population standard deviation. Round your answer to two decimal places.

31

Calculate the standard deviation for the following data represented by the frequency histogram. Round your answer to two decimal places.

32

Consider the set of scores displayed as a bar chart:

a

Create a cumulative frequency table for this data, with column titles: x, f, fx, and cf.

Hence calculate:

b

The median score.

c

The first quartile.

d

The third quartile.

e

The interquartile range.

Grouped data
33

For each of the following frequency tables:

i

Use the midpoint of each class interval to estimate the mean, correct to one decimal place.

ii

State the modal group of scores.

a
\text{Score }(x)\text{Frequency}
1 - 520
6-1015
11 - 158
16 - 204
21 - 253
26 - 302
b
\text{Score }(x)\text{Frequency}
0 \leq x < 204
20 \leq x < 4015
40 \leq x < 6023
60 \leq x < 8073
80 \leq x < 10045
34

Consider the following table:

a

Complete the table.

b

Calculate an estimate for the mean. Round your answer to two decimal places.

c

Calculate an estimate for the standard deviation. Round your answer to two decimal places.

d

If we used the original ungrouped data to calculate standard deviation, would we expect it to have a higher or lower standard deviation? Explain your answer.

\text{Class}\text{Class} \\ \text{centre } (x)ffx
1 - 98
10 - 186
19 - 274
28 - 366
37 - 458
\text{Total}
Box plots
35

For the box plot shown, find the interquartile range.

Scores
0
10
20
30
40
50
60
70
80
90
36

For the box plot shown, find each of the following:

a

The lowest score.

b

The highest score.

c

The range.

d

The median.

e

The interquartile range.

Scores
0
2
4
6
8
10
12
14
16
18
20
37

Consider the box plot shown:

a

Determine the percentage of scores that lie between the following:

i

7 and 15 inclusive

ii

1 and 7 inclusive

iii

19 and 9 inclusive

iv

7 and 19 inclusive

v

1 and 15 inclusive

b

In which quartile is the data the least spread out?

Scores
0
5
10
15
20
38

Create a box plot to represent the data in the given table:

\text{Minimum}5
Q_120
\text{Median}40
Q_355
\text{Maximum}70
39

The glass windows for an airplane are cut to a certain thickness, but machine production means there is some variation. The thickness of each pane of glass produced is measured (in millimetres) and the results are shown in the following dot plot:

a

Find the percentage of thicknesses between 10.8 mm and 11.2 mm inclusive, correct to two decimal places.

b

Determine the median thickness.

c

Calculate the interquartile range.

d

Construct a box plot to represent the data.

e

According to the box plot, in which quartile are the results the most spread out?

f

State whether the following can be determined from the box plot:

i

The mode thickness

ii

The frequency of each thickness

iii

The median thickness

iv

The spread of thicknesses

40

Two groups of people, athletes and non-athletes, had their resting heart rate (in beats per minute) measured. The results are displayed in the following pair of box plots.

a

Calculate the median heart rate of athletes.

b

Calculate the median heart rate of the non-athletes.

c

Which group has lower heart rates on average?

d

Calculate the interquartile range of the athletes' heart rates.

e

Calculate the interquartile range of the non-athletes' heart rates.

f

Which group has more consistent heart rate measures?

Athletes
40
50
60
70
80
90
Non-athletes
40
50
60
70
80
90
Outliers
41

The selling price of recently sold houses are given:

\$467\,000, \$413\,000, \$410\,000, \$456\,000, \$487\,000, \$929\,000

a

Calculate the mean selling price, rounded to the nearest thousand dollars.

b

Which of the prices raised the mean so that it is not reflective of most of the prices?

c

Recalculate the mean selling price excluding this outlier.

42

A set of data has a five-number summary as shown in the table:

a

Calculate the interquartile range.

A fence is a value 1.5 \times IQR above the upper quartile or below the lower quartile.

b

Calculate the value of the lower fence.

c

Calculate the value of the upper fence.

d

Hence determine if there is an outlier for this set of data.

Minimum5
Lower quartile6
Median12
Upper quartile17
Maximum28
43

VO_{2} Max is a measure of how efficiently your body uses oxygen during exercise. The more physically fit you are, the higher your VO_{2} Max. A group of people had their VO_{2} Max measured, the results are given below:

21,\text{ } 21,\text{ } 23,\text{ } 25,\text{ } 26,\text{ } 27,\text{ } 28,\text{ } 29,\text{ } 29,\text{ } 29,\text{ } 30,\text{ } 30,\text{ } 32,\text{ } 38,\text{ } 38,\text{ } 42,\text{ } 43,\text{ } 44,\text{ } 48,\text{ } 50,\text{ } 76

a

Determine the median VO_{2} Max.

b

Determine the upper quartile.

c

Determine the lower quartile.

d

Calculate 1.5 \times IQR, where IQR is the interquartile range. Round your answer to two decimal places.

e

An outlier is a score that is more than 1.5 \times IQR above or below the upper quartile or lower quartile respectively.

Determine if this set of data has any outliers and state their value if applicable.

f

Draw a box plot for this data, clearly indicating any outliers if applicable.

44

A group of Year 12 students were asked how many hours they spend on Hashtagram per day. The results are given below:

1.9, 1.1, \text{ }2.4, 2.3, \text{ }2.1, 1.2, \text{ }1.3, 1.6, \text{ }1.5, 1.8

a

Determine the five-number summary for this data set.

b

Another girl, Naylaa spends 3.6 hours using Hashtagram. If her score was added to this group, would it be considered an outlier? Explain your answer.

Shape of data
45

Describe the shape of each of the following data sets as positively skewed, negatively skewed or symmetrical:

a
b
c
d
e
Leaf
16 7 7
22 2 2 2 3 3 3
33 3 3 6 6 6 7 7 7 7 7
44 4 4 4 4 4
57 7

Key: 1 \vert 6 = 16

f
46

Describe the shape of the distribution for the following set of scores and corresponding box plot:

21,\text{ } 21,\text{ } 23,\text{ } 25,\text{ } 26,\text{ } 27,\text{ } 28,\text{ } 29,\text{ } 29,\text{ } 29,\text{ } 30,\text{ } 30,\text{ } 32,\text{ } 38,\text{ } 38,\text{ } 42,\text{ } 43,\text{ } 44,\text{ } 48,\text{ } 50,\text{ } 76
20
25
30
35
40
45
50
55
60
65
70
75
80
47

The following stem and leaf plot displays the ages of people who entered through the gates of a concert in the first 5 seconds:

a

Calculate the median age.

b

Find the difference between the lowest age and the median.

c

Find the difference between the highest age and the median.

d

Calculate the mean age, correct to two decimal places.

e

Describe the shape of the distribution.

Age
10 0 1 1 2 2 4 7 9
22 2 5 6 7
31 4 8
43
54

Key: 1 \vert 6 = 16

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Outcomes

ACMGM048

review the statistical investigation process; for example, identifying a problem and posing a statistical question, collecting or obtaining data, analysing the data, interpreting and communicating the results

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