Chapter 6Measures of Central Tendency 1

Chapter 6Measures of Central Tendency 1

Warm-up Exercise(page 6.1)

1. / (a) / Daily average temperatureC / Frequency
22-24 / 5
25-27 / 8
28-30 / 12
31-33 / 4
34-36 / 1

(b)

2. / (a) / Score less than / Cumulative frequency
20.5 / 0
30.5 / 1
40.5 / 2
50.5 / 7
60.5 / 19
70.5 / 26
80.5 / 32
90.5 / 36
100.5 / 40

(b)

Build-up Exercise

Exercise 6A

Elementary Set(page 6.2)

1.(a)Mean

(b)Mean

(c)Mean

2.(a)Mean

(b)Mean

(c)Mean

3.Mean selling price of the glasses

4.Mean weight of the oranges

5.Mean of the high jump records

1. Mean

7. / (a) / Datumx / Frequency(f) / fx
13 / 3 / 39
14 / 7 / 98
15 / 9 / 135
16 / 1 / 16
Total / 20 / 288

(b)Mean

8.Mean

9.Total travelling timeminutes

544minutes

10.Total timeminutes

75minutes

11.Mean



12.New mean

13.Mean price

14.Amount needed to pay

15.(a)Sum

(b)Mean

16. / (a) / Class / Class mark(x) / Frequency(f) / fx
26-30 / 28 / 3 / 84
31-35 / 33 / 9 / 297
36-40 / 38 / 15 / 570
41-45 / 43 / 8 / 344
46-50 / 48 / 5 / 240
Total / 40 / 1535

(b)Mean

Advanced Set(page 6.4)

1.(a)Mean

(b)Mean

(c)Mean

(d)Mean

(e)Mean

(f)Mean

2.Mean weight of the strawberries

1. Mean of their monthly salaries

4.(a)Mean

(b)Mean

(corr. to 3 sig. fig.)

(c)Mean

(corr. to 3 sig. fig.)

5. / Datumx / Frequency(f) / fx
10 / 10 / 100
15 / 15 / 225
20 / 30 / 600
25 / 20 / 500
30 / 10 / 300
35 / 10 / 350
40 / 5 / 200
Total / 100 / 2 275

Mean

6.Bill for lunch

7.Mean



8.New mean

(corr. to 3 sig. fig.)

9.Mean time

(corr. to 2 d.p.)

10.Mean electricity charge

11.Mean of the 45 numbers

12. / (a) / Timehour / Class mark
(x) / Frequency
(f) / fx
16-20 / 18 / 15 / 270
21-25 / 23 / 25 / 575
26-30 / 28 / 30 / 840
31-35 / 33 / 20 / 660
36-40 / 38 / 10 / 380
Total / 100 / 2 725

(b)Mean timehours

27.25hours

13.Let x be her score in the 6th test.

Mean score for the 6 tests

She should get 72.5 in the 6th test.

14.Let x be the number of boys and y be the number of girls in S3A.

Mean of the number of
the whole class /

There are 30 students.



Substituteinto (1),

Substituteinto (2),

There are 11 boys and 19 girls in S3A.

15. / Weightkg / 30-34 / 35-39 / 40-44 / 45-49 / 50-54
Frequency(f) / 8 / 12 / 12 / x / 2
Class markx / 32 / 37 / 42 / 47 / 52
fx / 256 / 444 / 504 / 47x / 104

Mean

16.Amount of water left after the accidenton the fifth day morning

Amount of water can be consumed by each member in each day

(corr. to 3 sig. fig.)

Exercise 6B

Elementary Set(page 6.6)

1.(a)Median

(b)Median

(c)Median

2.(a)Median

(b)Median

(c)Median

3.(a)Arrange the data in ascending order:

1, 2, 4, 5, 7, 8, 9

Median

(b)Arrange the data in ascending order:

10, 12, 13, 16, 17, 18, 20, 21, 24

Median

(c)Arrange the data in ascending order:

4.1, 2.9, 1.8, 1.4, 0.8, 0.9,

2.3, 3.0, 8.2

Median

4.(a)Arrange the data in ascending order:

45, 46, 46, 48, 49, 50, 53, 58

Median

(b)Arrange the data in ascending order:

8.9, 9.0, 9.0, 9.2, 9.5, 9.8, 9.8,

9.8, 10.1, 10.2

Median

(c)Arrange the data in ascending order:

40, 35, 28, 21, 27, 30, 32,

49, 57, 69

Median

5.Arrange the weights (in kg) of the ten pumpkins in ascending order:

Median weight of these ten pumpkins

6.[Total number of students49.Therefore, arrange the pocket money of these students in ascending order, the pocket money of the 25th student is the median pocket money.]

According to the table, arrange the pocket money in ascending order,

the pocket money of the 1st and the 2nd students is $3;

the pocket money from the 3rd to the 7th students is $4;

the pocket money from the 8th to the 17th students is $5;

the pocket money from the 18th to the 29th students is $6.

Median pocket money

7.[Total number of people50.Therefore, arrange the number of movies watched in ascending order, the median is the mean number of movies watched of the 25th and the 26th people.]

According to the table, arrange the number of movies watched in ascending order,

the number of movies watched from the 1st to the 4th people is 0;

the number of movies watched from the 5th to the 12th people is1;

the number of movies watched from the13th to the 24th people is 2;

the number of movies watched from the 25th to the 39th people is 3.

Median number of movies watched

8.[Total frequency100.Therefore, the median I.Q. test result is the value corresponding to the cumulative frequency 50.]

From the graph,

median I.Q. test result

9.[Total frequency347.Therefore, arrangethe data in ascending order, the median is the 4th datum.]

According to the diagram,

median

10.(a)Total number of students

(b)[Total number of students32.Therefore, arrange the test scores in ascending order, the median is the mean test score of the 16th and the 17th students.]

From the diagram,

median test score

11.Arrange the data in ascending order:

x1, x1, x2, x3, x5

Median

12.(a)Arrange the quantity (in glasses) in ascending order:

103, 106, 106, 110, 113, 121

Median

(b)Arrange the quantity (in glasses) in ascending order:

103, 106, 106, 110, 113, 121, 135

Median

13.[Total frequency40.Therefore, arrange the time spent in ascending order, the median is the mean time spent of the 20th and the 21st students.]

According to the table, arrange the time spent in ascending order,

the time spent from the 1st to the 5th students is 0 hour;

the time spent from the 6th to the 12th students is 1 hour;

the time spent from the 13th to the 20th students is 2 hours;

the time spent from the 21st to the 30th students is 3 hours.

Median time spenthours

2.5hours

14.Median14

15.Let x1, xandx1 be the three consecutive numbers, then the median is x.

Mean24

The median of the three numbers is 24.

Advanced Set(page 6.9)

1.(a)Median

(b)Median

(c)Median

(d)Median

2.(a)Arrange the data in ascending order:

3, 4, 5, 10, 12, 14, 20

Median

(b)Arrange the data in ascending order:

Median

(c)Arrange the data in ascending order:

4, 5, 5.5, 6, 6, 6.5, 7, 7, 8, 8, 8, 9

Median

(d)Arrange the data in ascending order:

3, 2.8, 0.9, 0.3, 0.4, 1.2, 1.4, 2.1, 2.3, 2.4

Median

3.Arrange the weights (in kg) of the eight watermelons in ascending order:

4.2, 4.6, 4.9, 4.9, 5.0, 5.1, 5.3, 5.4

Median weight

4.[Total frequency85.Therefore, arrange the number of phone surveys received in ascending order, the median is the number of phone surveys received by the 43rd people.]

According to the table, arrange the number of phone surveys received in ascending order,

the number of phone surveys received from the 1st to the 31st people is 0;

the number of phone surveys received from the 32nd to the 53rd people is 1.

Median number of phone surveys received

5.[Total frequency40.Therefore, the median weight is the value corresponding to the cumulative frequency 20.]

From the graph, median weight

6.[Totalfrequency475319.Therefore, arrange the data in ascending order, the median is the 10th datum.]

From the diagram, median

7.Arrange the data in ascending order:

2x7, 2x7, 2x5, 2x3, 2x1, 2x

Median

8.(a)Total number of students

(b)[Total frequency50.Therefore, arrange the number of pairs of sport shoes in ascending order, the median is the mean number of pairs of sport shoes of the 25th and the 26th students.]

According to the table, arrange the number of pairs of sport shoes in ascending order,

the number of pairs of sport shoes from the 1st to the 4th students is 1;

the number of pairs of sport shoes from the 5th to the 25th students is 2;

the number of pairs of sport shoes from the 26th to the 49th students is 3.

Median number of pairs of sport shoes owned by these students

9.Median

10.Let x1, x, x1 and x2 be the fourconsecutive numbers, then

median

Mean17.5

The median of the four numbers is 17.5.

11.The median of the (2n1) consecutive numbers is the(n1)th number.

Since the (2n1) numbers are consecutive numbers,the mean of the (2n1) numbers is the(n1)th number.

i.e. MedianMean

12.(a)Arrange the moving service quotes (in $) of the ten moving companies in ascending order:

800, 1000, 1100, 1200, 1200, 1250, 1300, 1300, 1400, 1500

Median quote

(b)Arrange the moving service quotes (in $) of the ten moving companies in ascending order:

1000, 1100, 1200, 1200, 1250, 1300, 1300, 1400, 1500, 1800

Actual median quote

13.(a)(i)Mean

(ii)Arrange the expenditures (in $) in ascending order:

54, 82, 91, 125, 163

Median expenditure of the five purchases

(b)Arrange the expenditures (in $) in ascending order:

54, 82, 91, 102, 125, 163

Median expenditure of the six purchases

14.(a)

(b)From the cumulative frequency polygon in(a),

median duration for holding breath

15.(a)[Total frequency30.Therefore, arrange the weights in ascending order, the median is the mean of the weights of the 15th and the 16th chickens.]

From the diagram,

median weight of chickens from farm A

(b)[Total frequency30.Therefore, arrange the weights in ascending order, the median is the mean of the weights of the 15th and the 16th chickens.]

From the diagram,

median weight of chickens from farm B

(c)(i)

(ii)[Total frequency60.Therefore, arrange the weights in ascending order, the median is the mean of the weights of the 30th and the 31st chickens.]

From the diagram in(c)(i),

median weight of 60 chickens from farms A and B

Exercise 6C

Elementary Set(page 6.12)

1.(a)[The datum10 occurs 4 times and has the highest frequency.]

Mode

(b)[The datum9 occurs 3 times and has the highest frequency.]

Mode

(c)[Both the data21 and 26 occur twice and have the highest frequency.]

Modes21 and 26

2.(a)Arrange the data in ascending order:

3, 3, 3, 4, 5, 6, 6, 8, 8

[The datum3 occurs 3 times and has the highest frequency.]

Mode

(b)Arrange the data in ascending order:

13, 13, 9, 8, 5, 5, 5, 0

[The datum5 occurs 3 times and has the highest frequency.]

Mode

(c)Arrange the data in ascending order:

1.6, 1.7, 1.7, 1.9, 2.0, 2.1, 2.3, 2.3

[Both the data1.7 and 2.3 occur twice and have the highest frequency.]

Modes1.7 and 2.3

3.[Most families have 1 child.]

Mode of the number of children

4.[Most students live in public housing.]

Mode of the type of housing that students

live in is public housing.

5.(a)[The interval36-40 has the highest frequency.]

Modal class36-40

(b)[Both the intervals7-9 and 13-15 have the highest frequency.]

Modal classes7-9 and 13-15

6.The mode of the sales volume of fruit juice is orange juice.

7. / (a) / Time(minute) / Frequency
11-20 / 4
21-30 / 10
31-40 / 22
41-50 / 4

(b)[The class interval 31minutes-40minutes has the highest frequency.]

Modal class of travelling time to school

31minutes-40minutes

8.Angle at centre corresponding to the folding type

The folding type occupies the largest angle at the centre.

The mode of the types of umbrellas sold

is the folding type.

9.[According to the diagram, the datum4 occurs 3 times and has the highest frequency.]

Mode of the sales volume

10. / (a) / Working hours(xhours) / Frequency
/ 5
/ 0
/ 2
/ 2
/ 7
/ 8
/ 3
/ 2
/ 1

(b)[The class interval 9hours-10hours has the highest frequency.]

Modal class of working hours yesterday

9 hours-10 hours

11.(a)From the graph,

(b) / Electricity charge for
a certain period($) / Frequency
301-400 / 2
401-500 / 4
501-600 / 6
601-700 / 22
701-800 / 30
801-900 / 20
901-1000 / 6

(c)[The class interval $701-$800 has the highest frequency.]

Modal class of electricity charges

(d)Frequency of the class interval

$701-$800

Advanced Set(page 6.16)

1.(a)[The datum12 occurs 3 times and has the highest frequency.]

Mode

(b)[The data2, 0 and 1 occur twice and have the highest frequency.]

Modes2, 0 and 1

(c)[The data,andoccur twice and have the highest frequency.]

Modes,and

2.(a)Arrange the data in ascending order:

149, 149, 149, 150, 150, 150, 154, 162

[Both the data149 and 150 occur 3 times and have the highest frequency.]

Modes149 and 150

(b)Arrange the data in ascending order:

0, 0, 0, 3, 3, 5, 8, 8, 9, 14

[The datum0 occurs 3 times and has the highest frequency.]

Mode

(c)Arrange the data in ascending order:

5, 9, 10, 11, 13, 17, 19

All the data occur once only.

The set of data does not have a mode.

3.[Both the shoes sizesand 7 have the highest frequency.]

The modes of the shoes sizes areand 7.

4.[The intervals21-30 and 51-60 have the highest frequency.]

The modal classes are 21-30 and 51-60.

5.(a)[The angle at the centre corresponding to ‘3’ is the greatest.]

The mode number of magazines bought

weekly is 3.

(b)Number of citizens who buy 3 magazines weekly

6.[The frequencies of ages 13 and 14 are the highest.]

The modes of the ages of students started

wearing glasses are 13 and 14.

7.[From the graph, the class interval 100 minutes-109 minutes has the highest frequency.]

The modal class of the duration of the

movies is 100 minutes-109 minutes.

8.(a)[For the boys, most students have 1 subject with credit. For the girls, most students have four subjects with credits.]

The mode of number of subjects with

credits for boys is 1.

The mode of number of subjects with

credits for girls is 4.

(b) / Number of subjects with credits / Frequency for all secondary three students
0 / 21
1 / 30
2 / 24
3 / 21
4 / 25
5 / 32
6 / 21
7 / 12
8 / 7

[Most students have 5 subjects with credits.]

The mode of number of subjects with

credits for all secondary three students

is 5.

9.(a)[The age 22 occurs 6 times and has the highest frequency.]

The mode of the ages is 22.

(b) / (i) / Age / Frequency
18-22 / 13
23-27 / 16
28-32 / 9
33-37 / 5
38-42 / 4
43-47 / 3

(ii)[The class interval 23-27 has the highest frequency.]

The modal class of ages is23-27.

10. / (a) / Floor area(m2) / Frequency
10-29 / 3
30-49 / 2
50-69 / 20
70-89 / 5

(b)[The class interval 50m2-69m2 has the highest frequency.]

The modal class of floor areas is

-.

11.(a)From the graph,

the number of students in the group

is 60.

(b) / Time spent on chatting on the internet (hour) / Frequency
5.1-10.0 / 3
10.1-15.0 / 7
15.1-20.0 / 14
20.1-25.0 / 23
25.1-30.0 / 10
30.1-35.0 / 3

(c)[The class interval 20.1 hours-25.0 hours has the highest frequency.]

The modal class of time spent on

chatting on the internet is

20.1 hours-25.0 hours.

(d)The frequency of the class interval 20.1 hours-25.0 hours

Exercise 6D

Elementary Set(page 6.20)

1.(a)Arrange the time (in hour) in ascending order:

12, 20, 20, 21, 23, 24, 24, 24, 28, 30

Meanhours

22.6 hours

Medianhours

23.5 hours

Mode24 hours

(b)Holly requires more than 22.6 hours and less than 24 hours to knit a scarf.

Time required by her23hours

2.(a)Arrange the revenues (in ten thousand dollars) in ascending order:

103, 104, 105, 106, 107, 112, 115,118, 119, 236

Mean of the revenues

Median of the revenues

Each revenue occurs once only.

The revenues do not have a mode.

(b)Arrange the new revenues (in ten thousand dollars) in ascending order:

104, 105, 106, 107, 112, 115,

118, 119

New mean of the revenues

New median of the revenues

Each revenue occurs once only.

The revenues do not have a mode.

3.(a)Arrange the data in ascending order:

23, 24, 24, 24, 24, 30, 32, 37, 56, 60,

72, 72, 75, 75, 80, 80, 81, 81, 81, 84,

84, 85, 85, 85, 87, 87, 92, 92, 95, 98

[The data24 occurs 4 times and has the highest frequency.]

Mode

(b)Suggested answer:

No, because there are 5 data less than or equal to 24, and the remaining 25 data are greater than 24 in which 22 of them is two times of 24. Therefore, the mode cannot reflect the real situation.

4.(a)Arrange the salaries (in $) in ascending order:

5000,6000,6000,6000,7000, 7000,8000,8000,9000,10000, 10000,10000,11000,11000, 12000, 50000

Mean of the salaries

Median of the salaries

Modes of the salaries$6000 and $10000

(b)The median can reflect the real situation.

5.(a)No, because $12 (the mode) is the lowest price of all items, but only 3 items are selling at a price of $12. Therefore, I do not agree with the owner.

(b)Yes, because the stock of items which are selling at a price of $12 each occupies 70% of all stock. Therefore, I agree with what the owner claims.

Advanced Set(page 6.21)

1.(a)Arrange the time (in month) in ascending order:

9, 9, 10, 10, 10, 11, 11, 12, 12, 12,

12, 13, 13, 13, 14, 14, 15, 15, 15, 16

Mean of learning time

months

12.3months

Median of learning timemonths

12months

Mode of learning time12months

(b)Since the mean, median and mode are close to each other, all of them can reflect the real situation.

2.Arrange the monthly net profits (in $) in ascending order:

27000, 30000, 30000, 31000, 33000,

34000, 34000, 35000, 42000, 43000,

45000, 46000

Mean

(corr. to 1 d.p.)

Median

Modes$30000 and $34000

Since the mean is higher than the median and the modes, the owner of the toy shop possibly used the mean to support his statement.

3.(a)(i)Arrange the number of shots hitting the targets by Mr. Chan in ascending order:

1, 2, 4, 4, 5, 5, 6, 7

Mean of the number of shots hitting the targets by Mr. Chan

Median of the number of shots hitting the targets by Mr. Chan

Modes of the number of shots hitting the targets by Mr. Chan4 and 5

(ii)Arrange the number of shots hitting the targets by Mr. Lee in ascending order:

1, 2, 3, 3, 5, 6, 7, 8

Mean of the number of shots hitting the targets by Mr. Lee

Median of the number of shots hitting the targets by Mr. Lee

Mode of the number of shots hitting the targets by Mr. Lee

(b)Since in the 8 rounds, Mr. Chan won in 3 rounds, Mr. Lee won in 4 rounds and they drew in one round. Therefore, Mr. Lee is leading in most of the rounds.

(c)The mean of the number of shots hitting the targetsby Mr. Lee is greater than that by Mr. Chan. But the median and the modesof the number of shots hitting the targetsby Mr. Lee are less than that by Mr. Chan. Therefore, the mean reflects the reality better.

4.(a)Price of the wallpaper in the original offer

Price of the wallpaper in the special offer

The price of the special offer is higher than that of the original offer.

Mr. Chan should not accept the

special offer.

(b)Amount required

Mean price of the wallpaper

5.(a)Mean of the original prices

Median of the original prices

The original prices do not have mode.

Mean of the new prices

Median of the new prices

The new prices do not have mode.

The manufacturer comes to the

conclusion by using the median.

(b)The mean can better reflect the situation.

Difference between the means of the original prices and the new prices

Exercise 6E

Elementary Set(page 6.22)

1.(a)Mean

Median

Mode

(b)Each new datum is obtained by adding 2kg to the original datum.

New meanOriginal mean2kg

New medianOriginal median2kg

New modeOriginal mode2kg

2.(a)Mean

Median

Mode

(b)Each new datum is obtained by subtracting 5mL from the original datum.

New meanOriginal mean5mL

New medianOriginal median5mL

New modeOriginal mode5mL

3.(a)Mean

Median

Mode

(b) / Mean / Median / Mode
Each datum is multiplied by 3 / 21 / 21 / 21
Each datum is multiplied by / 3.5 / 3.5 / 3.5

4.Each new datum is obtained by adding x to the original datum.

New medianOriginal medianx

5.Each new datum is obtained by subtracting 3 from the original datum.

New modeOriginal mode3

21Original mode3

Original mode

6.(a)Arrange the data in ascending order:

102, 110, 113, 115, 115, 117,

117, 118, 118, 118, 119, 119,

120, 129, 148

Median

(b)Arrange the data in ascending order:

102, 110, 113, 115, 115, 117,

117, 118, 118, 118, 119, 119,

119, 120, 120, 129, 148

New median

(c)Arrange the data in ascending order:

102, 110, 113, 115, 115, 117, 117, 118, 118, 118, 119, 119, 120, 129

New median

7.(a)Sum of this set of data

(b)(i)Sum of the remaining data

(ii)New mean

8.(a)Mode

(b)Each new datum is obtained by multiplying x to the original datum.

New modeOriginal modex

9.(a)Mean

(b)New mean

(c)New mean

10.(a)Let x be the score of each question.

The score obtained in the test is the number of correct answers times the score of each question.

 Mean of the scores

Mean of the number of

correctanswersx



The score of each question is 2.

(b)The score obtained in the test is the number of correct answers times the score of each question.

Median of the scores

Median of the number of

correctanswers2

(c)The score obtained in the test is the number of correct answers times the score of each question.

Mode of the scores

Mode of the number of correct

answers2

34Mode of the number of correct

answers2

Mode of the number of correct answers

Advanced Set(page 6.24)

1.(a)Mean