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

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P. 2 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. Example: Find the average height of a 10-year-old boy: Mathematics in Workplaces 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. 137 cm

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P 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. Table 6.2 Lower class limit Upper class limit 12//// //// //20 – 24 10//// 15 – 19 5////10 – //// //// ////5 – 9 FrequencyTallyTime spent (min) 12 min Class mark

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P. 4 Class width Upper class boundary – Lower class boundary 6.1 Organization of Continuous Data (14.5 – 9.5) min 5 min 12//// //// //20 – 24 10//// 15 – 19 5////10 – //// //// ////5 – 9 FrequencyTallyTime spent (min)

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P. 5 Example 6.1T (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 The following frequency distribution table shows the average speed (in km/h) of some vehicles passing through a tunnel. (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 Frequency 75–7970–7465–6960–6455–5950–5445–49Average Speed (km/h) 6.1 Organization of Continuous Data Solution:

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P. 6 The following shows the results of the high jump (in m) for 40 sportsmen. (a)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. 6.1 Organization of Continuous Data Example 6.2T Solution: (b)Class width (1.595 – 1.545) m (a)Click here: Frequency distribution table (c)Required percentage 30% 0.05 m

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P Histogram, Frequency Polygon and Frequency Curve The following table shows the heights (in cm) of 145 plants. Table 6.16 We can construct a histogram for the data. Height (cm)Class mark (cm)Class boundaries (cm)Frequency 20 – – – – – – – – – – – – A. Histogram

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P. 8 Height (cm)Class mark (cm)Class boundaries (cm)Frequency 20 – – Table 6.16 Fig. 6.5 A. Histogram 6.2 Histogram, Frequency Polygon and Frequency Curve 30 – – – – – – – – – –

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P. 9 The following shows the mobile phone monthly usage (in minutes) of some businessmen – – – – – – – – – – – – 399 FrequencyClass boundaries (in minutes)Usage (minutes) Construct a histogram for the above data. A. Histogram 6.2 Histogram, Frequency Polygon and Frequency Curve Example 6.3T

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P. 10 A. Histogram 6.2 Histogram, Frequency Polygon and Frequency Curve Example 6.3T Solution:

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P. 11 A histogram looks similar to a bar chart. However, there are some differences between them. Table 6.19 A. Histogram 6.2 Histogram, Frequency Polygon and Frequency Curve NoYesWith gaps between bars For continuous dataFor discrete dataNature of data Histogram Bar chart

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P. 12 Using the histogram in Fig. 6.5, we can obtain a frequency polygon. The total area of the bars in a histogram should be equal to the area under its corresponding frequency polygon. Fig Histogram, Frequency Polygon and Frequency Curve B. Frequency Polygon

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P. 13 The following table shows the areas (in cm 2 ) of 50 tiles in different shapes – – – – – – 40 FrequencyClass mark (cm 2 )Area (cm 2 ) Construct a frequency polygon for the above data. B. Frequency Polygon 6.2 Histogram, Frequency Polygon and Frequency Curve Example 6.4T

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P. 14 B. Frequency Polygon 6.2 Histogram, Frequency Polygon and Frequency Curve Example 6.4T Solution:

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P. 15 The following frequency polygon shows the average daily working time (in hours) of some doctors. (a)Construct a frequency distribution table from the frequency polygon. (b)Which class interval has the most doctors? (c)How many doctors work more than 10 hours a day? (d)Find the number of doctors interviewed for the above frequency polygon. B. Frequency Polygon 6.2 Histogram, Frequency Polygon and Frequency Curve Example 6.5T Solution: (b)The class interval 10 hours – 11 hours. (c) Number of doctors work more than 10 hours (d)Number of doctors Frequency distribution table

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P. 16 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 The curve does not necessarily pass through all the vertices of the frequency polygon. 6.2 Histogram, Frequency Polygon and Frequency Curve

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P. 17 The following graph shows the test marks of Chinese and English for S.2A. (a)In which subject do most students get marks higher than 80? (b)In which subject do most students get marks lower that 60? (c)Which subject is performed better? (b)Chinese (c)English, as the mean mark of English is greater than Chinese. C. Frequency Curve 6.2 Histogram, Frequency Polygon and Frequency Curve (a)English Solution: Example 6.6T

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P Cumulative Frequency Polygon and Cumulative Frequency Curve Table 6.28 A. Cumulative Frequency Polygon The following shows the English test marks earned by 40 students. Marks Class boundaries Number of students 40 – – – – – – – – – – – – Marks less than Number of students Cumulative frequency tableFrequency distribution table Table 6.29

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P. 19 Fig the cumulative frequency must start from zero, 3. the last point on the cumulative frequency polygon refers to the total number of data. A. Cumulative Frequency Polygon 6.3Cumulative Frequency Polygon and Cumulative Frequency Curve 1. the trend of the graph never goes down since the cumulative frequency never decreases, From the cumulative frequency polygon, we can see that

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P. 20 The cumulative frequency polygon shows the marks of a Mathematics test for S.2A. (a)How many students are there in S.2A? (b)Find the percentage of students whose marks lie between 60 – 80 marks. (c)If the passing mark is 50, find the percentage of students who failed the test. A. Cumulative Frequency Polygon 6.3Cumulative Frequency Polygon and Cumulative Frequency Curve Solution: (a) There are 30 students. (b) Required percentage (c) Required percentage Example 6.7T

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P. 21 B. Cumulative Frequency Curve Fig Cumulative Frequency Polygon and Cumulative Frequency Curve Like a frequency curve, we can smooth a cumulative frequency polygon to become a curve called a cumulative frequency curve.

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P. 22 B. Cumulative Frequency Curve Like cumulative frequency polygon, 2. the cumulative frequency must start from zero, 3. the last point on the cumulative frequency curve refers to the total number of data. 6.3Cumulative Frequency Polygon and Cumulative Frequency Curve 1. the trend of the graph never goes down since the cumulative frequency never decreases, Fig. 6.26

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P. 23 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? B. Cumulative Frequency Curve 6.3Cumulative Frequency Polygon and Cumulative Frequency Curve Example 6.8T Solution: (a) There are 40 drinks. (b) 16 drinks contain less than 6% alcohol. (c) Number of drinks

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P. 24 We can obtain the corresponding data from a particular number of observations by using percentiles and quartiles. Fig Cumulative Frequency Polygon and Cumulative Frequency Curve C. Percentiles, Quartiles and Median

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P. 25 C. Percentiles, Quartiles and Median 6.3Cumulative Frequency Polygon and Cumulative Frequency Curve 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. Solution: (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. Example 6.9T

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

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P. 27 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. (a)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. 6.4 Abuses of Statistics Example 6.10T 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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P. 28 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? (b)Suggest a way to reduce the misunderstanding from the graph. 6.4 Abuses of Statistics Example 6.11T 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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P. 29 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. (b)Suppose that Mark and Ada spent the same amount on food. Find the ratio of their total expenditures. 6.4 Abuses of Statistics Example 6.12T 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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P. 30 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. 4.Class boundaries are the extreme values in each class interval. 3.

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P Histogram, Frequency Polygon and Frequency Curve Chapter Summary 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. A.Histogram 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. B.Frequency Polygon and Frequency Curve

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P Cumulative Frequency Polygon and Cumulative Frequency Curve Chapter Summary 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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P. 33 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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P Abuses of Statistics Chapter Summary 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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6.1 Organization of Continuous Data Follow-up 6.1 The following frequency distribution table shows the average daily amount of money spent on lunch by the staff in a company. (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 Frequency 60 – 6950 – 5940 – 4930 – 3920 – 2910 – 19Amount ($) Solution: (a)Lower class limit $10, upper class limit $19 (b)Class width $(19.5 – 9.5) $10 (c)The class $40 – $49 has the highest frequency. Lower class boundary $9.5, upper class boundary $19.5 Class mark $(49 40) 2 $44.5

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The following shows the results of a 100 m race (in s) for 30 students. (a) Complete the frequency distribution table below. (b)What is the class width? (c)Find the percentage of students who finished the race within s. 6.1 Organization of Continuous Data Follow-up Time (s)Class Mark (s)Class boundaries (s)Frequency 12.0 – 13.9 Solution: Time (s)Class Mark (s)Class boundaries (s)Frequency 12.0 – – – – – – (c)Required percentage(b)Class width (13.95 – 11.95) s 2 s

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The following table shows the highest daily temperatures (in C) of city A in August – – – – – – – – – – – – 26.0 Frequency Class boundaries ( C)Temperature ( C) Table 6.18 Construct a histogram for the data. A. Histogram 6.2 Histogram, Frequency Polygon and Frequency Curve Follow-up 6.3

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A. Histogram 6.2 Histogram, Frequency Polygon and Frequency Curve Follow-up 6.3 Solution:

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B. Frequency Polygon 6.2 Histogram, Frequency Polygon and Frequency Curve The following table shows the distribution of the length of hand spans (in cm) of 30 students – – – – – – 14.0 FrequencyClass mark (cm)Hand span (cm) Construct a frequency polygon for the data. Table 6.21 Follow-up 6.4

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B. Frequency Polygon 6.2 Histogram, Frequency Polygon and Frequency Curve Follow-up 6.4 Solution:

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B. Frequency Polygon 6.2 Histogram, Frequency Polygon and Frequency Curve The frequency polygon shows the long jump records (in m) of some sportsmen. Fig (a)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? Follow-up 6.5

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B. Frequency Polygon 6.2 Histogram, Frequency Polygon and Frequency Curve (b) The class interval 4.45 m – 4.95 m has the most sportsmen. (c) sportsmen jump a distance less than 3.95 m. (d) Total number of sportsmen 72. FrequencyClass mark (m)Distance (m) (a) – 3.45 Follow-up 6.5 Solution: – – – – – 5.95

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The 2 frequency curves show the weights (in kg) of schoolbags of boys and girls in a class. (a)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? (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. 6.2 Histogram, Frequency Polygon and Frequency Curve C. Frequency Curve Follow-up 6.6 Solution:

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A. Cumulative Frequency Polygon 6.3Cumulative Frequency Polygon and Cumulative Frequency Curve The cumulative frequency polygon shows the weights (in pounds) of puppies in a pet shop. (a)How many puppies are there in the pet shop? (b)How many puppies weigh between 8 pounds and 13 pounds? (c)What is the percentage of puppies that weigh more than 23 pounds? Follow-up 6.7 (a) There are 20 puppies. Solution: (b) Number of puppies 10 – 4 (c) Required percentage 6

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B. Cumulative Frequency Curve The cumulative frequency curve shows the blood pressures (in mm/Hg) of some patients. (a)How many patients are there? (b)How many patients have blood pressures between 120 mm/Hg and 140 mm/Hg? 6.3Cumulative Frequency Polygon and Cumulative Frequency Curve Follow-up 6.8 Solution: (a)There are 38 patients. (b) Number of patients 35 – 8 27

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C. Percentiles, Quartiles and Median The cumulative frequency polygon shows the distances (in m) of 20 discus throws made by a sportsman. (a)Find the 30th percentile. (b)Find the 80th percentile. 6.3Cumulative Frequency Polygon and Cumulative Frequency Curve Follow-up 6.9 Solution: (a)Corresponding cumulative frequency for the 30th percentile 20 30% 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. 6 16

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6.4 Abuses of Statistics (a)What is the ratio of the heights of these 3 bars?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? (c)Does the diagram mislead readers? Explain your answer.Does the diagram mislead readers? Explain your answer. The bar chart shows the numbers of crimes in a city in the last 3 months. Follow-up 6.10 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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6.4 Abuses of Statistics The graph shows the number of As obtained by the students from Super Tutorial Centre and Smart Tutorial Centre in a public examination. (a)Mable thinks that the number of As 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. Follow-up 6.11 Solution: (a)No. The actual numbers of As 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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6.4 Abuses of Statistics Follow-up 6.11 The graph shows the number of As 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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6.4 Abuses of Statistics Comment on the following statements: (a)The number of comic books in both stores are the same.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. The pie charts show the distribution of books in Bookstores A and B. Follow-up 6.12 Solution: Since we dont 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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The following shows the results of the high jump (in m) for 40 sportsmen. (a)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. Example 6.2T Organization of Continuous Data Solution: (a) – – – – – – – – – – – – 1.59 FrequencyClass boundaries (m)Class mark (m)Height (m)

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The following frequency polygon shows the average daily working time (in hours) of some doctors. (a)Construct a frequency distribution table from the frequency polygon. B. Frequency Polygon 6.2 Histogram, Frequency Polygon and Frequency Curve Example 6.5T Solution: – – 11 9 – 10 8 – 9 7 – 8 6 – 7 Frequency Class Mark (hours) Working Time (hours)

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