中三數學教學筆記 第六章 集中趨勢的量度 章節 標題 教學筆記 家課

6.1 引言 - Students should be able to tell details from the graphical presentations of statistics.

- Students should be able to summarize the meanings and express in terms of the measure of central tendency.

6.2 平均數 (arithmetic mean) - Students should be able to evaluate the arithmetic mean. P.196 CW 6.3 & 6.4 Ex.6A Q.7-10, 14

6.3 中位數 (median) - Students should be able to evaluate the median. - Students should be able to tell the importance of median in statistics

involving salaries.

Ex.6B Q.1, 6, 8

6.4 眾數和眾數組 (mode / modal class) - Students should be able to evaluate the mode and the modal class. Ex.6C Q.1, 4, 5 6.5 集中趨勢量度的應用 - Students should be able to identify the different applications for the

measures of central tendency.

6.6 誤用平均值 - Students should be able to general abuse of central tendency in statistics. 6.7 集中趨勢量度的探索 - Students should be able to solve problems involving the measures of central

tendency.

6.8 加權平均數 (weighted mean) - Students should be able to evaluate the weighted mean. Ex.6F Q.1, 5 備註：學生應懂得利用計算機的 "SD mode”來計算平均數的算題。 參考書籍： 1. 生活的數學，羅浩源編著：「平均何所指﹖」、「半數兒童高於平均體重﹖」、「從數學中可以找到公平﹖」、「怎樣量度生活消費」、「打

字機或電腦鍵盤上的字母為何不作順序排列﹖」、「如何檢驗統計圖的誤用﹖」

仁愛堂田家炳中學 Handout 6-1 中三級 數學科：第六章 集中趨勢的量度

平均何所指﹖ 在資訊不斷膨脹的今天，我們經常有機會接觸到與統計有關的信息。由於平均

數﹙average﹚具有總結大量數據的簡便效用，因此屢見於報紙和其他傳播媒介的報

道上。其實，平均數﹙average﹚可分為三大類： 1. 算術平均數﹙arithmetic mean﹚、 2. 中位數﹙median﹚及 3. 眾數﹙mode﹚。

很可惜，一般的報道並沒有清楚說明，祇讓讀者各自去理解。以下圖為例：

你會認為該新聞特寫標題上的「人均收入」是指哪一種平均﹖ 在決定採用哪種平均時，統計者須考慮每種平均本身的特點是否適合於某處境

的描述上。一般而言，有關收入的統計若以中位數作平均的，可說是較為公平的做

法。無論該處的財富收入怎樣分佈。中位數總會讓我們知道半數人口的收入低於何

等水平。在財富收入不勻的社會裡，其餘兩種平均最好留好勞資雙方討價還價之

用。當資方以平均工資偏高﹙少數特高薪金可將算術平均值拉高﹗﹚為理據去凍結

加薪時，勞方則可用眾數作為平均來說明「勞苦大眾」的平均工資還是偏低的。 在日常生活裡，平均意念正廣泛地使用。加強對各種平均數的認識，應有助我

們掌握更準確的信息。

半數兒童高於平均體重﹖

當小甘參考上面的統圖時，竟發現自己的身高和體重交匯點比中位數﹙median﹚高出 1.2 倍。媽媽趁此機會勸誡小甘勿偏吃過量的肉食。不料小甘卻這樣回應說：「不

要緊，以我的高度來說，反正有一半人和我一樣高於平均體重呢﹗」 由於該份調查報告採用中位數作為顯示身高及體重比例的平均指標，小甘的說

法是正確的。但小甘還是最好聽媽媽的忠告，盡量減低進食脂肪肉類的分量。 其實，從統計的角度看，以中位數表達體型的平均意念，最大的好處是可免受

過高或偏低的數據影響，破壞它的代表作用。

想一想 為何以中位數去制定

男女體型的標準﹖

想一想 標題上的「人均收入」

是指哪一種平均﹖

怎樣量度生活消費﹖

以物價指數量度生活消費是統計學應用於日常生活的典型例子之一。在經濟學

上，物價指數可分為甲類、乙類和恆生三種。而受到一般市民所關注的則是甲類消

費物價指數，因為普羅大眾的消費意欲可從其中反映出來。但在統計學上，物價指

數是基於甚麼原理計算出來的呢﹖ 其實，物價指數是量度某一段時間內物價的平均改變程度。

在圖中的「甲類消費物價指數籃子」，把日常的開支歸納為九大類型，而每類已設

定的消費比重可反映我們支出這些消費上的負擔，則可用消費比重 (W) 加權於物價

相對變動比率 (P1)，再加以加權平均數﹙weighted mean﹚的計算方法把物價指數找

出來。

根據上表，物價指數 = Σ(P1 × W)

ΣW = 110.1﹙準確至 1 位小數﹚。

換言之，相對基年期來說，甲類物價指數上升了約 10 點。

摘自【生活的數學】，羅浩源編著 想一想 物價指數怎樣反映 生活的平均消費﹖

消費種類 比重 (W)(%)

物價升幅

(%)*

物價相對變動比率 P1 = (1 + 升幅%) × 100

P1 × W

燃料及電力 3.37 15 115 387.55 食 品 37.30 10 110 4103.00 交 通 7.17 18 118 846.06 雜項物品 6.03 8 108 651.24 住 屋 25.34 9 109 2762.06雜項服務 9.27 8 108 1001.16 煙 酒 2.06 20 120 247.20 耐用物品 4.34 7 107 464.38 衣 履 5.12 6 106 542.72

ΣW = 100 Σ(P1 × W) = 11005.37

YAN OI TONG TIN KA PING SECONDARY SCHOOL Handout 6-2

F.3 Mathematics: Measures of Central Tendencies Histogram: Frequency Distribution Table: The heights of 80 people

(a) For the class 160 - 164, the lower class limit is cm. the upper class limit is cm. (b) For the class 165 - 169, the lower class boundary is cm. the upper class boundary is cm. (c) The class mark for 170 - 174 is cm. (d) The class width is cm. (e) The mean height is cm. The median height is cm. The modal class is .

(f) Draw the frequency polygon in the graph provided. The heights of 80 people

(g) Draw the Cumulative Frequency Polygon in the graph below. The heights of 80 people

160 164 159.5 164.5 172 5 168.625 169.5

170 cm - 174 cm 147 152 157 162 167 172 177 182 187

Height (cm)

Cum

ulat

ive

Freq

uenc

y

80

60

40

20

0

Table for the heights of 80 people

Height (cm) Frequency 150 - 154

155 - 159

160 - 164

165 - 169

170 - 174

175 - 179

180 - 184

4

6

10

20

24

12

4

Freq

uenc

y

24

20

16

12

8

4

0

147 152 157 162 167 172 177 182 187

Height (cm)

149.5 154.5 159.5 164.5 169.5 174.5 179.5 184.5

Height (cm)

Freq

uenc

y 24

20

16

12

8

4

0

YAN OI TONG TIN KA PING SECONDARY SCHOOL Handout 6-3(a)

F.3 Mathematics: Measures of Central Tendencies 1. Identify the types of the following graphs of statistics. A. B. C. D. E. F. G. 2. Given 8 numbers: 2, 2, 2, 3, 3, 4, 5 and 7. What is the mean of them? What is the mode of them? What is the median of them? 3. For the set of numbers: 2, x, 12, 3, x and 11, the mean equals to x. The median of the numbers is . 4. The mean of (a - 2), (b + 3) and (c + 5) is 6. The mean of (a + 4), (b + 6) and (c - 1) is . 5. In a class of 42 students, the mean height is 1.62 m. If the mean height of 18

students is 1.54 m, the mean height of the rest is

m. 6. The mean of a, b, c, d is 4, and the mean of e, f, g, h, i, j is 6. The mean of all ten

numbers is .

7. The mean of a, b and c is 15; the mean of b, c and d is 20; and the mean of a, b, c and d is 16. The mean of b and c is .

8. The scores of 40 students in a certain test are given in the following table: The average score of them is . 9. The number of fishes caught by a man in eight days are 17, 14, 16, 17, 14, 14, 18

and 17. The median number of fishes caught is . The difference of the mean and the median is . The mode of fishes caught is . 10. Given a group of numbers. When one of the numbers is increased by 10, the mean

increases by 1. How many numbers are there in the group? 11. The table shows the mean marks of two classes of students in a Mathematics test: A student in Class A has scored 91 marks. It is found that his score was wrongly

recorded as 19 in the calculation of the mean mark for Class A in the above table. Find the correct mean mark of the 80 students in the classes.

Marks 50 - 59 60 - 69 70 - 79 80 - 89 90 - 99 Number of students 6 12 14 5 3

Number of students Mean mark Class A 38 72Class B 42 54

20.5

broken line graph histogram frequency polygon cumulative (折線圖) (直方圖) (頻數多邊形) frequency polygon

(累積頻數多邊形) 71.25

bar chart (棒形圖) pie chart (圓形圖) box plot (盒須圖) 16.5 3.5 0.625 2 14 and 17 3 10 7 7 1.68 63.45 5.2

12. The mean marks of the students in A, B and C in a test are given in the table below: If the overall mean mark of these 3 classes is 60.7, x = . 13. The following table shows the weekly pocket money (in dollars) received by 30

students. (a) The mean pocket money received by the students is $ . (b) If the weekly pocket money of 4 students are each increased from $70 to $90,

the new mean weekly pocket money received by the students is $ , correct to the nearest $0.1. 14. The table below show the time (in minutes) that 40 Form 3 students need to return

home after school.

If the mean time that these 40 students need to return home after school is 11.2 minutes, x = ,

y = .

15. The following histogram shows the weights of a group of boys.

The Class Width of the histogram is kg. The Upper Class Boundary of the Fourth Class is kg. The Lower Class Boundary of the Second Class is kg. The Class Mark of the Third Class is kg. The total number of boys in the group is . The mean weight of them is kg. The median weight is kg. The modal class for them is . 16. 以下是一位小學生的考試成績，但其中兩科的成績被墨水弄污了：

試求出該兩科的成嫧。

Time (min) 10 11 12 13 14 15 16Number of students 20 6 5 x y 1 0

科目 中文 英文 數學 自然 社會 健教 平均 分數 89 74 6 . . 8 81 85 79

52.5 kg - 57.5 kg 67 及 78 分 5 3

Class A B CNumber of students 42 40 38

Mean mark 60 64 x

20

15

10

5

0

Freq

uenc

y

50 55 60 65

Weight (kg)

Weekly pocket money ($) 50 60 70 80 90 100 Number of students 2 4 9 8 6 1

58 5 67.5 52.5 60 50 58.5

75

57.5 77.7

Handout 6-3(b)

17. Consider the following figure, find (a) the Class Mark of the First Class, (b) the Lower Class Boundary of the Second Class, (c) the Mean, (d) the Median, and (e) the Modal Class. 18. Consider the following figure, find

(a) the Class Mark of the First Class, (b) the Lower Class Boundary of the Second Class, (c) the Mean, (d) the Median, and (e) the Modal Class. 19. The following figure shows the cumulative frequency of 400 students in a

Mathematics contest.

(a) From the cumulative frequency polygon, (1) The mean is ____________________________. (2) The First Quartile is ______________________. (3) The Median is __________________________. (4) The Third Quartile is _____________________. (5) The Interquartile Range is _________________. (b) A student with marks greater than or equal to 50 will be awarded a prize. (1) The number of students who will be awarded prizes is __________________.

(2) If one student is chosen at random from these 400 students, the probability that the student is a prize winner is _________________________________.

f.

100

80

20

030 40 50 60 70

Length (cm)

c.f.

100

80

20

030 40 50 60

Length (cm)

c.f.

400

300

200

100

0

10 20 30 40 50 60

Marks

35 cm 40 cm 45 cm 45 cm 40 cm - 50 cm 40 cm 45 cm 54 cm 55 cm 55 cm - 65 cm

32.5 10 40 50 40 100

0.25

Chapter 6 Measures of Central Tendencies Quiz 6-1 F.3______ Name:_____________________________________( ) Marks:_____/ 21 1. Ten years ago, the mean age of a band of 11 musicians

was 30. One of them is now leaving the band at the age of 40. What is the present mean age of the remaining 10 musicians?

A. 40 B. 39 C. 37 D. 30 E. 29 2. The mean weight of 36 boys and 32 girls is 46 kg. If

the mean weight of the boys is 52 kg, then the mean weight of the girls is

A. 39.25 kg B. 40 kg C. 40.67 kg D. 49 kg 3. The mean of a set of 9 numbers is 12. If the mean of

the first 5 numbers is 8, the sum of the other four numbers is

A. 16 B. 40 C. 64 D. 68 E. 100

4. The median of the five numbers 15, x - 1, x - 3, x – 4 and x + 17 is 8. Find the mean of the five numbers.

A. 8 B. 12 C. 13.6 D. 14.4 5. If the mean of five numbers 15, x + 4, x + 1, 2x - 7 and

x - 3 is 6, then the mode of the five numbers is A. 1 B. 4 C. 5 D. 15 6. For the five numbers x, x – 1, x – 2, x, x + 8, which of the

following must be true? The median is x – 2 The mean is x + 1 The mode is 2 A. only B. only C. and only D. and only

7. The mean of (a + 4), (b + 5), (c – 3) is 6. The mean of (a – 7), (b + 13), (c – 3) is

A. 5 B. 6 C. 7 D. 8 8. If the mean of the numbers 3, 3, 3, 3, 4, 4, 5, 5, 6, x

is also x, which of the following is/are true? Mean = Mode Mode = Median Median = Mean A. only B. only C. only D. All of them

9. Consider the following test marks and their

corresponding weighting (比重／加權). Test 1 Test 2 Test 3

Test marks 65 70 90Weighting 25% 35% 40%

Find their weighted mean. A. 75 B. 76.75 C. 78.21 D. 90

10. The histogram in the Figure shows the distribution of scores of a class of 40 students in a test.

(a) Complete the Table.

Frequency distribution table for the scores of 40 students

Score (x) Class mid-value

(Class mark 組中數)Frequency

頻數

44 < x < 52 3

52 < x < 60

64 15

68 < x < 76 11

80

(3 marks)

(b) Estimate (估算) the mean (算術平均數), modal class ( 眾數組 ) and median ( 中位數 ) of the distribution. (3 marks)

11. The marks scored by eleven students in a mathematics

quiz are as follows: 10 20 30 45 50 60 65 65 65 70 70 Find (1) the mean, (2) the mode (眾數), and (3) the median

of the above marks. (3 marks)

12. The following cumulative frequency polygon (累積頻數多邊形) shows the age distribution of the workers in a factory.

(a) How many workers are there in the factory? (b) Estimate the median age of the workers.

(3 marks) (a) From the c.f.polygon, there are 100 workers

in the factory. (b) For the 100 workers, median age is the age

of the 50th worker. At c.f. value 50, draw a horizontal line and intersect the c.f.polygon, then drop a vertical line to the axis.

Median age = 31.5 years old

Mean = 64 marks

Modal class = 60 marks to 68 marks

Median = (60 + 8 × 815 ) marks

= 64415 marks (≈ 64.27 marks)

(1) Mean = 50 marks

(2) Mode = 65 marks

(3) Median = 60 marks

48 56 9 60 < x < 68

72

76 < x < 84 2

Chapter 6 Measures of Central Tendencies Quiz 6-2 F.3______ Name:_____________________________________( ) Marks:_____/ 23 1. Find the mean, mode and median for the set of data:

4, 4, 5, 8, 8, 10, 10, 10, 15 and 18. (3 marks)

2. A set of 10 numbers has a mean of 25. If a number 3 is

added, find the new mean. (2 marks)

3. The table below shows the number of students in 3 classes of a school and their marks in a mathematics test.

Class Number of student Average marks F.3A 40 61 F.3B x 70 F.3C 35 50

If the overall average marks of the three classes is 60, find x. (3 marks)

4. The mean of a, b, c is 3, and the mean of d and e is 8. What is the mean of a, b, c, d and e? (2 marks)

5. If the mean of (a+1), (b+2), (c-3) and d is 7, find the mean of (a-1), (b-3), c and (d-4). (3 marks)

6. If the mean of 4, 6, 9, 4, 6 and (x – 1) is 5.5, find

the mode of these numbers. (3 marks)

(40 + x + 35)60 = 61 × 40 + 70x + 50 × 35 x = 31

Mean = 3 × 3 + 8 × 2

3 + 2

= 5

Given (a+1) + (b+2) + (c-3) + d = 7 × 4

then (a-1) + (b-3) + c + (d-4) = 20

the mean of the four new numbers is 5.

4 + 6 + 9 + 4 + 6 + (x – 1) = 5.5 × 6

x = 5

ie. the numbers are 4, 4, 4, 6, 6 and 9.

Hence, mode of these numbers is 4.

Mean = 4 + 4 +5 + 8 + 8 + 10 + 10 + 10 + 15 + 18

10

= 9.2 Mode = 10 Median = 9

Mean = 25 × 10 + 3

10 + 1

= 23

7. The following cumulative frequency polygon shows the distribution of the weights of 30 students. (7 marks)

(a) Write down the median weight of the students. (b) Fill in the following table.

Weight (kg) Class Mark (kg) Frequency

40-44 42 5

45-49

50-54

55-59

60-64

(c) Find the mean weight of the students. (d) Find the modal class of the students.

Mean weight

= 42 × 5 + 47 × 5 + 52 × 10 + 57 × 8 + 62 × 2

30 kg

= 51.5 kg Modal class = 50 kg - 54 kg

Median weight = 52 kg

47 5

52 10

57 8

62 2