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**Construction and Interpretation of Simple Diagrams and Graphs (II)**

6 Construction and Interpretation of Simple Diagrams and Graphs (II) Mathematics in Workplaces 6.1 Organization of Continuous Data 6.2 Histogram, Frequency Polygon and Frequency Curve 6.3 Cumulative Frequency Polygon and Cumulative Frequency Curve 6.4 Abuses of Statistics Chapter Summary

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**Mathematics in Workplaces**

Nutritionist Growth charts give information on the standard measures concerning weight for age, height for age, weight for height, etc. By using charts, a nutritionist can give suggestions to patients concerning the diet. 137 cm Example: Find the average height of a 10-year-old boy: As shown in the figure, construct a vertical line for age 10 to meet the curve of 50%, then construct a horizontal line to meet the vertical axis.

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**6.1 Organization of Continuous Data**

The following is a typical frequency distribution table showing the daily time spent (in minutes) on reading newspapers in a class of students. 12 //// //// // 20 – 24 10 //// //// 15 – 19 5 //// 10 – 14 14 //// //// //// 5 – 9 Frequency Tally Time spent (min) Class mark Lower class limit Upper class limit Table 6.2 12 min

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**6.1 Organization of Continuous Data**

12 //// //// // 20 – 24 10 //// //// 15 – 19 5 //// 10 – 14 14 //// //// //// 5 – 9 Frequency Tally Time spent (min) Class width Upper class boundary – Lower class boundary (14.5 – 9.5) min 5 min

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**Example 6.1T 6.1 Organization of Continuous Data Solution:**

The following frequency distribution table shows the average speed (in km/h) of some vehicles passing through a tunnel. 24 36 17 20 18 15 5 Frequency 75–79 70–74 65–69 60–64 55–59 50–54 45–49 Average Speed (km/h) (a) Find the class limits of the 2nd class. (b) Find the class boundaries of the class with the highest frequency. (c) Find the class mark of the last class. (d) Find the class width. Solution: (a) Lower class limit 50 km/h, upper class limit 54 km/h (b) 69.5 km/h – 74.5 km/h (c) Class mark (79 75) km/h 2 77 km/h (d) Class width (74.5 – 69.5) km/h 5 km/h

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**Example 6.2T 6.1 Organization of Continuous Data Solution:**

The following shows the results of the high jump (in m) for 40 sportsmen. Using the above data, construct a frequency distribution table with the first 2 classes ‘1.55 m – 1.59 m’ and ‘1.60 m – 1.64 m’ including the class mark, class boundaries and frequency of each class. (b) Find the class width. (c) Find the percentage of sportsmen who can jump higher than m. Solution: (a) Click here: Frequency distribution table (b) Class width = (1.595 – 1.545) m (c) Required percentage = 30% = 0.05 m

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**6.2 Histogram, Frequency Polygon and Frequency Curve**

A. Histogram The following table shows the heights (in cm) of 145 plants. Height (cm) Class mark (cm) Class boundaries (cm) Frequency 20 – 29 24.5 19.5 – 29.5 15 30 – 39 34.5 29.5 – 39.5 25 40 – 49 44.5 39.5 – 49.5 30 50 – 59 54.5 49.5 – 59.5 35 60 – 69 64.5 59.5 – 69.5 70 – 79 74.5 69.5 – 79.5 Table 6.16 We can construct a histogram for the data.

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**6.2 Histogram, Frequency Polygon and Frequency Curve**

A. Histogram Height (cm) Class mark (cm) Class boundaries (cm) Frequency 20 – 29 24.5 19.5 – 29.5 15 30 – 39 34.5 29.5 – 39.5 25 40 – 49 44.5 39.5 – 49.5 30 50 – 59 54.5 49.5 – 59.5 35 60 – 69 64.5 59.5 – 69.5 25 70 – 79 74.5 69.5 – 79.5 15 Table 6.16 Fig. 6.5

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**Class boundaries (in minutes)**

6.2 Histogram, Frequency Polygon and Frequency Curve A. Histogram Example 6.3T The following shows the mobile phone monthly usage (in minutes) of some businessmen. 5 – 1600 – 1899 12 – 1300 – 1599 35 999.5 – 1000 – 1299 28 699.5 – 999.5 700 – 999 23 399.5 – 699.5 400 – 699 10 99.5 – 399.5 100 – 399 Frequency Class boundaries (in minutes) Usage (minutes) Construct a histogram for the above data.

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**Example 6.3T 6.2 Histogram, Frequency Polygon and Frequency Curve**

A. Histogram Example 6.3T Solution:

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**6.2 Histogram, Frequency Polygon and Frequency Curve**

A. Histogram A histogram looks similar to a bar chart. However, there are some differences between them. No Yes With gaps between bars For continuous data For discrete data Nature of data Histogram Bar chart Table 6.19

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**6.2 Histogram, Frequency Polygon and Frequency Curve**

B. Frequency Polygon Using the histogram in Fig. 6.5, we can obtain a frequency polygon. Fig. 6.30 The total area of the bars in a histogram should be equal to the area under its corresponding frequency polygon.

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**Example 6.4T 6.2 Histogram, Frequency Polygon and Frequency Curve**

B. Frequency Polygon Example 6.4T The following table shows the areas (in cm2) of 50 tiles in different shapes. 15 130.5 121 – 140 8 110.5 101 – 120 6 90.5 81 – 100 3 70.5 61 – 80 50.5 41 – 60 12 30.5 21 – 40 Frequency Class mark (cm2) Area (cm2) Construct a frequency polygon for the above data.

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**Example 6.4T 6.2 Histogram, Frequency Polygon and Frequency Curve**

B. Frequency Polygon Example 6.4T Solution:

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**Example 6.5T 6.2 Histogram, Frequency Polygon and Frequency Curve**

B. Frequency Polygon Example 6.5T The following frequency polygon shows the average daily working time (in hours) of some doctors. Construct a frequency distribution table from the frequency polygon. (b) Which class interval has the most doctors? How many doctors work more than 10 hours a day? (d) Find the number of doctors interviewed for the above frequency polygon. 10 13 20 22 35 17 Solution: Frequency distribution table The class interval ‘10 hours – 11 hours’. (c) Number of doctors work more than 10 hours = 35 17 = 52 Number of doctors = 10 13 20 22 35 17 = 117

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**6.2 Histogram, Frequency Polygon and Frequency Curve**

C. Frequency Curve Referring to Fig. 6.7, we can smooth the frequency polygon to become a curve called a frequency curve for the distribution. Fig. 6.11 The curve does not necessarily pass through all the vertices of the frequency polygon.

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**Example 6.6T 6.2 Histogram, Frequency Polygon and Frequency Curve**

C. Frequency Curve Example 6.6T The following graph shows the test marks of Chinese and English for S.2A. In which subject do most students get marks higher than 80? In which subject do most students get marks lower that 60? (c) Which subject is performed better? Solution: (a) English (b) Chinese (c) English, as the mean mark of English is greater than Chinese.

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**6.3 Cumulative Frequency Polygon and Cumulative Frequency Curve**

A. Cumulative Frequency Polygon The following shows the English test marks earned by 40 students. Frequency distribution table Cumulative frequency table Marks less than Number of students 39.5 Marks Class boundaries Number of students 40 – 49 39.5 – 49.5 5 50 – 59 49.5 – 59.5 6 60 – 69 59.5 – 69.5 8 70 – 79 69.5 – 79.5 13 80 – 89 79.5 – 89.5 90 – 99 89.5 – 99.5 3 49.5 5 59.5 5 + 6 11 69.5 19 79.5 32 89.5 37 99.5 40 Table 6.28 Table 6.29

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**6.3 Cumulative Frequency Polygon and Cumulative Frequency Curve**

A. Cumulative Frequency Polygon From the cumulative frequency polygon, we can see that 1. the trend of the graph never goes down since the cumulative frequency never decreases, 2. the cumulative frequency must start from zero, 3. the last point on the cumulative frequency polygon refers to the total number of data. Fig. 6.23

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**6.3 Cumulative Frequency Polygon and Cumulative Frequency Curve**

A. Cumulative Frequency Polygon Example 6.7T The cumulative frequency polygon shows the marks of a Mathematics test for S.2A. How many students are there in S.2A? Find the percentage of students whose marks lie between 60 – 80 marks. If the passing mark is 50, find the percentage of students who failed the test. Solution: (a) There are 30 students. (c) Required percentage (b) Required percentage

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**B. Cumulative Frequency Curve**

6.3 Cumulative Frequency Polygon and Cumulative Frequency Curve B. Cumulative Frequency Curve Like a frequency curve, we can smooth a cumulative frequency polygon to become a curve called a cumulative frequency curve. Fig. 6.26

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**B. Cumulative Frequency Curve**

6.3 Cumulative Frequency Polygon and Cumulative Frequency Curve B. Cumulative Frequency Curve Like cumulative frequency polygon, 1. the trend of the graph never goes down since the cumulative frequency never decreases, 2. the cumulative frequency must start from zero, 3. the last point on the cumulative frequency curve refers to the total number of data. Fig. 6.26

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**6.3 Cumulative Frequency Polygon and Cumulative Frequency Curve**

B. Cumulative Frequency Curve Example 6.8T The following cumulative frequency curve shows the alcoholic concentration of some drinks. (a) How many drinks are there? (b) How many drinks contain less than 6% alcohol? (c) How many drinks have more than 7% alcohol? Solution: (a) There are 40 drinks. (b) 16 drinks contain less than 6% alcohol. (c) Number of drinks 40 28 12

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**6.3 Cumulative Frequency Polygon and Cumulative Frequency Curve**

C. Percentiles, Quartiles and Median We can obtain the corresponding data from a particular number of observations by using percentiles and quartiles. Fig. 6.30

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**6.3 Cumulative Frequency Polygon and Cumulative Frequency Curve**

C. Percentiles, Quartiles and Median Example 6.9T The following cumulative frequency curve shows the lengths (in cm) of some springs. (a) Find the total number of springs. (b) What are the lower quartile and upper quartile? (c) Find the 50th percentile. 24 16 Solution: 8 (a) There are 32 springs. (b) Corresponding cumulative frequency for the lower quartile 8 and the upper quartile 24. From the graph, the lower quartile 7 cm and the upper quartile 15.4 cm. (c) Corresponding cumulative frequency for the 50th percentile 16. From the graph, 50th percentile 11.4 cm.

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6.4 Abuses of Statistics Statistical diagrams and graphs are helpful in presenting data. They assist in interpreting the information collected. However, we may be easily misled by statistical graphs if we do not study and analyze the graphs carefully.

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**Example 6.10T 6.4 Abuses of Statistics Solution:**

The bar chart shows the passing rates of S.2 A, S.2 B, S.2 C and S.2 D in the Mathematics examination. Find the ratio of the heights of the last two bars. (b) What is the ratio of the passing rates of S.2 C and S.2 D? (c) Does the diagram mislead readers? Explain your answer briefly. Solution: (a) Ratio of the heights of the last two bars 3 : 7 (b) Ratio of the passing rates 45 : 65 9 : 13 (c) Yes. The ratio of the heights of the bars is different from the ratio of the actual passing rates, thus the graph mislead readers.

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**Example 6.11T 6.4 Abuses of Statistics Solution:**

The graph shows the number of readers who read fiction and non-fiction in a library at a particular time. (a) How does the graph mislead people? Suggest a way to reduce the misunderstanding from the graph. Solution: (a) The difference in the widths of the figures exaggerates the ratio of the actual number of readers in fiction and non-fiction. (b) Redraw the diagrams with the same width.

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**Example 6.12T 6.4 Abuses of Statistics Solution:**

The pie charts show the monthly expenditures of Mark and Ada. (a) Comment on the following statement. ‘The amount Mark spent on travel is the same as the amount Ada spent on food.’ Suppose that Mark and Ada spent the same amount on food. Find the ratio of their total expenditures. Solution: (a) Their actual monthly expenditures are not given. So the amount Mark spent on travel may not be the same as Ada. (b) Since they spent the same amount on food, required ratio

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**Chapter Summary 6.1 Organization of Continuous Data**

1. Continuous data can be organized in a frequency distribution table. 2. Class limits are the end points of each class interval. 3. 4. Class boundaries are the extreme values in each class interval.

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**6.2 Histogram, Frequency Polygon and Frequency Curve**

Chapter Summary 6.2 Histogram, Frequency Polygon and Frequency Curve A. Histogram 1. The horizontal axis can be either the class boundaries or the class marks. 2. The vertical axis shows the frequencies. 3. Data are represented by rectangular bars with no gaps between the bars. B. Frequency Polygon and Frequency Curve In a histogram, if we add an extra class interval with zero frequency on both ends of the distribution and mark the class mark on the top of each bar, we can obtain 1. a frequency polygon by joining the class marks with line segments; 2. a frequency curve by smoothing the frequency polygon.

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**6.3 Cumulative Frequency Polygon and Cumulative Frequency Curve**

Chapter Summary 6.3 Cumulative Frequency Polygon and Cumulative Frequency Curve A. Cumulative Frequency Polygon and Cumulative Frequency Curve If we want to study the overall distribution, we con construct the cumulative frequency polygon/curve from a cumulative frequency table. In a cumulative frequency polygon or a cumulative frequency curve, 1. the trend of the graph never goes down since the cumulative frequency never decreases; 2. the cumulative frequency must start from zero; 3. the last point on the cumulative frequency polygon/curve refers to the total number of data.

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Chapter Summary 6.3 Cumulative Frequency Polygon and Cumulative Frequency Curve B. Percentiles, Quartiles and Median 1. The pth percentile of a set of data is the number such that p percent of the data is less than that number. 2. If we divide the distribution into 4 equal quartiles, then the 25th percentile, 50th percentile and 75th percentile are called the lower quartile, the median, and the upper quartile respectively.

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**Chapter Summary 6.4 Abuses of Statistics**

When interpreting a statistical diagram, we should pay attention to the following: 1. the scale of the axes of the graph, 2. the sizes of the figures, 3. the actual frequencies of data.

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**Follow-up 6.1 6.1 Organization of Continuous Data Solution:**

The following frequency distribution table shows the average daily amount of money spent on lunch by the staff in a company. 2 6 13 12 5 Frequency 60 – 69 50 – 59 40 – 49 30 – 39 20 – 29 10 – 19 Amount ($) (a) Find the class limits and class boundaries of the 1st class. (b) Find the class width. (c) Find the class mark of the class with the highest frequency. Solution: (a) Lower class limit $10, upper class limit $19 Lower class boundary $9.5, upper class boundary $19.5 (b) Class width $(19.5 – 9.5) $10 (c) The class ‘$40 – $49’ has the highest frequency. Class mark $(49 40) 2 $44.5

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**Follow-up 6.2 6.1 Organization of Continuous Data Solution:**

The following shows the results of a 100 m race (in s) for 30 students. (a) Complete the frequency distribution table below. Time (s) Class Mark (s) Class boundaries (s) Frequency 12.0 – 13.9 12.95 11.95 – 13.95 11 14.0 – 15.9 14.95 13.95 – 15.95 15 16.0 – 17.9 16.95 15.95 – 17.95 4 Time (s) Class Mark (s) Class boundaries (s) Frequency 12.0 – 13.9 (b) What is the class width? Find the percentage of students who finished the race within s. Solution: (b) Class width = (13.95 – 11.95) s (c) Required percentage = 2 s

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**Follow-up 6.3 6.2 Histogram, Frequency Polygon and Frequency Curve**

A. Histogram Follow-up 6.3 The following table shows the highest daily temperatures (in C) of city A in August. 3 28.05 – 28.55 28.1 – 28.5 8 27.55 – 28.05 27.6 – 28.0 6 27.05 – 27.55 27.1 – 27.5 7 26.55 – 27.05 26.6 – 27.0 4 26.05 – 26.55 26.1 – 26.5 25.55 – 26.05 25.6 – 26.0 Frequency Class boundaries (C) Temperature (C) Table 6.18 Construct a histogram for the data.

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**Follow-up 6.3 6.2 Histogram, Frequency Polygon and Frequency Curve**

A. Histogram Follow-up 6.3 Solution:

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**Follow-up 6.4 6.2 Histogram, Frequency Polygon and Frequency Curve**

B. Frequency Polygon Follow-up 6.4 The following table shows the distribution of the length of hand spans (in cm) of 30 students. 4 16.3 16.1 – 16.5 7 15.8 15.6 – 16.0 15.3 15.1 – 15.5 6 14.8 14.6 – 15.0 14.3 14.1 – 14.5 2 13.8 13.6 – 14.0 Frequency Class mark (cm) Hand span (cm) Table 6.21 Construct a frequency polygon for the data.

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**Follow-up 6.4 6.2 Histogram, Frequency Polygon and Frequency Curve**

B. Frequency Polygon Follow-up 6.4 Solution:

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**Follow-up 6.5 6.2 Histogram, Frequency Polygon and Frequency Curve**

B. Frequency Polygon Follow-up 6.5 The frequency polygon shows the long jump records (in m) of some sportsmen. Fig. 6.10 Construct a frequency distribution table from the frequency polygon. (b) Which class interval has the most sportsmen? (c) How many sportsmen jump a distance less than 3.95 m? (d) How many sportsmen are there?

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**Follow-up 6.5 6.2 Histogram, Frequency Polygon and Frequency Curve**

B. Frequency Polygon Follow-up 6.5 Solution: Frequency Class mark (m) Distance (m) (a) 6 3.2 2.95 – 3.45 8 3.7 3.45 – 3.95 16 4.2 3.95 – 4.45 26 4.7 4.45 – 4.95 12 5.2 4.95 – 5.45 4 5.7 5.45 – 5.95 (b) The class interval ‘4.45 m – 4.95 m’ has the most sportsmen. (c) 6 8 = 14 sportsmen jump a distance less than 3.95 m. (d) Total number of sportsmen = 72.

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**Follow-up 6.6 6.2 Histogram, Frequency Polygon and Frequency Curve**

C. Frequency Curve Follow-up 6.6 The 2 frequency curves show the weights (in kg) of schoolbags of boys and girls in a class. Whose schoolbags do not have weights heavier than 5 kg? (b) Whose schoolbags do not have weights lighter than 3 kg? (c) Whose schoolbags are heavier in general? Solution: (a) Girls (b) Boys (c) The frequency curve of boys lies to the right of the curve of girls, thus the schoolbags of boys are heavier in general.

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**6.3 Cumulative Frequency Polygon and Cumulative Frequency Curve**

A. Cumulative Frequency Polygon Follow-up 6.7 The cumulative frequency polygon shows the weights (in pounds) of puppies in a pet shop. How many puppies are there in the pet shop? (b) How many puppies weigh between 8 pounds and 13 pounds? What is the percentage of puppies that weigh more than 23 pounds? Solution: (a) There are 20 puppies. (c) Required percentage (b) Number of puppies 10 – 4 6

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**6.3 Cumulative Frequency Polygon and Cumulative Frequency Curve**

B. Cumulative Frequency Curve Follow-up 6.8 The cumulative frequency curve shows the blood pressures (in mm/Hg) of some patients. How many patients are there? How many patients have blood pressures between 120 mm/Hg and 140 mm/Hg? Solution: (a) There are 38 patients. (b) Number of patients 35 – 8 27

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**6.3 Cumulative Frequency Polygon and Cumulative Frequency Curve**

C. Percentiles, Quartiles and Median Follow-up 6.9 The cumulative frequency polygon shows the distances (in m) of 20 discus throws made by a sportsman. Find the 30th percentile. (b) Find the 80th percentile. 16 Solution: (a) Corresponding cumulative frequency for the 30th percentile 20 30% 6. 6 From the graph, the 30th percentile 63.7 m. (b) Corresponding cumulative frequency for the 80th percentile 20 80% 16. From the graph, the 80th percentile 70.6 m.

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**Follow-up 6.10 6.4 Abuses of Statistics Solution:**

The bar chart shows the numbers of crimes in a city in the last 3 months. What is the ratio of the heights of these 3 bars? (b) What is the ratio of the number of crimes of these 3 months? Does the diagram mislead readers? Explain your answer. Solution: (a) Ratio of the heights 3 : 2 : 1 (b) Ratio of the number of crimes 45 : 40 : 35 9 : 8 : 7 (c) As the ratio of the height of the crime is different from the ratio of the actual number of crimes in a city in the last 3 months, the diagram mislead leaders.

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**Follow-up 6.11 6.4 Abuses of Statistics Solution:**

The graph shows the number of A’s obtained by the students from Super Tutorial Centre and Smart Tutorial Centre in a public examination. Mable thinks that the number of A’s obtained by the students from Super Tutorial Centre is about 6 times of that obtained by the students from Smart Tutorial Centre. Do you agree? Explain your answer briefly. Solution: (a) No. The actual numbers of A’s obtained by the students from Super Tutorial Centre and Smart Tutorial Centre in a public examination are 60 and 30 respectively.

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**Follow-up 6.11 6.4 Abuses of Statistics Solution:**

The graph shows the number of A’s obtained by the students from Super Tutorial Centre and Smart Tutorial Centre in a public examination. (b) How does the graph mislead readers? (c) Suggest a way to reduce the misunderstanding caused by the graph. Solution: (b) (1) The ratio of the areas of the diagrams is different from the actual results. (2) The statement claims that Super Tutorial Centre is the best. (c) Redraw the diagrams with the same width.

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**Follow-up 6.12 6.4 Abuses of Statistics Solution:**

The pie charts show the distribution of books in Bookstores A and B. Comment on the following statements: The number of comic books in both stores are the same. (b) The number of textbooks in bookstore B is larger than that in bookstore A. Solution: Since we don’t know the total number of books in each store, we cannot compare the number of different kinds of books between the 2 stores.

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**Example 6.2T 6.1 Organization of Continuous Data Solution:**

The following shows the results of the high jump (in m) for 40 sportsmen. Using the above data, construct a frequency distribution table with the first 2 classes ‘1.55 m – 1.59 m’ and ‘1.60 m – 1.64 m’ including the class mark, class boundaries and frequency of each class. Solution: (a) 1.80 – 1.84 1.75 – 1.79 1.70 – 1.74 1.65 – 1.69 1.60 – 1.64 1.55 – 1.59 Frequency Class boundaries (m) Class mark (m) Height (m) 1.82 1.77 1.72 1.67 1.62 1.57 1.795 – 1.845 1.745 – 1.795 1.695 – 1.745 1.645 – 1.695 1.595 – 1.645 1.545 – 1.595 8 4 2 10 7 9

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**Example 6.5T 6.2 Histogram, Frequency Polygon and Frequency Curve**

B. Frequency Polygon Example 6.5T The following frequency polygon shows the average daily working time (in hours) of some doctors. Construct a frequency distribution table from the frequency polygon. 10 13 20 22 35 17 Solution: Frequency Class Mark (hours) Working Time (hours) 11 – 12 10 – 11 9 – 10 8 – 9 7 – 8 6 – 7 11.5 10.5 9.5 8.5 7.5 6.5 17 35 22 20 13 10

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