6 More about Statistical Diagrams and Graphs 6.1 Organization of Continuous Data 6.2 Histograms, Frequency Polygons and Frequency Curves 6.3 Cumulative Frequency Polygons and Cumulative Frequency Curves 6.4 Abuses of Statistics

6.1 Organization of Continuous Data

6.1 Organization of Continuous Data

6.1 Organization of Continuous Data

6.1 Organization of Continuous Data

Example 1T Solution: 6 More about Statistical Diagrams and Graphs 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 of the 2nd class Upper class limit of the 2nd class

Example 1T Solution: 6 More about Statistical Diagrams and Graphs 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: (b) The 6th class has the highest frequency. Lower class boundary of the 6th class Upper class boundary of the 6th class

Example 1T Solution: 6 More about Statistical Diagrams and Graphs 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: (c) Class mark of the last class (d) Class width  (74.5 – 69.5) km/h

Example 2T 6 More about Statistical Diagrams and Graphs The following shows the results of the high jump (in metres) for 40 sportsmen. 1.81 1.55 1.69 1.59 1.74 1.79 1.60 1.75 1.73 1.56 1.66 1.58 1.78 1.59 1.67 1.68 1.71 1.72 1.80 1.65 1.73 1.70 1.61 1.77 1.68 1.69 1.57 1.66 1.75 1.58 1.74 1.76 1.75 1.77 1.59 1.78 1.76 1.64 1.62 1.69 Using the above data, construct a frequency distribution table with ‘1.55 m – 1.59 m’ and ‘1.60 m – 1.64 m’ as the first two class intervals, showing 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 1.745 m.

Example 2T Solution: 6 More about Statistical Diagrams and Graphs (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 (b) Class width = (1.595 – 1.545) m (c) Required percentage = 0.05 m

6.2 Histograms, Frequency Polygons and Frequency Curves

6.2 Histograms, Frequency Polygons and Frequency Curves A. Histograms

6.2 Histograms, Frequency Polygons and Frequency Curves A. Histograms

Example 3T Solution: 6 More about Statistical Diagrams and Graphs Refer to the figure. (a) Find the range of the heights that has the highest frequency. (b) Find the number of plants that are shorter than 49.5 cm. (c) Find the difference in the number of plants between the highest and the lowest frequencies. (d) Find the greatest possible difference in the heights of the tallest plant and the shortest plants. Solution: (a) The required range is ‘49.5 cm  59.5 cm’. (b) Number of plants shorter than 49.5 cm  15  25  30

Example 3T Solution: 6 More about Statistical Diagrams and Graphs Refer to the figure. (a) Find the range of the heights that has the highest frequency. (b) Find the number of plants that are shorter than 49.5 cm. (c) Find the difference in the number of plants between the highest and the lowest frequencies. (d) Find the greatest possible difference in the heights of the tallest plant and the shortest plants. Solution: (c) The difference  35  15 (d) The greatest possible difference  (79.5  19.5) cm

Class boundaries (in minutes) 6 More about Statistical Diagrams and Graphs Example 4T The following shows the mobile phone monthly usage (in minutes) of some businessmen. 5 1599.5 – 1899.5 1600 – 1899 12 1299.5 – 1599.5 1300 – 1599 35 999.5 – 1299.5 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. Find the class mark of the class interval represented by the tallest bar in the histogram.

The mobile phone monthly usage of some businessmen 6 More about Statistical Diagrams and Graphs Example 4T The mobile phone monthly usage of some businessmen Solution: (a) (b) From the graph, the 4th bar is the tallest. The required class mark

6.2 Histograms, Frequency Polygons and Frequency Curves B. Frequency Polygons extra class interval

Example 5T 6 More about Statistical Diagrams and Graphs 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.

Example 5T Solution: 6 More about Statistical Diagrams and Graphs The areas of 50 tiles in different shapes

Example 6T 6 More about Statistical Diagrams and Graphs The following frequency polygon shows the average daily working time (in hours) of some doctors. The class ‘11 hours – 12 hours’ has the highest frequency. Construct a frequency distribution table from the frequency polygon, showing the class interval, the class mark and the frequency of each class. (b) What percentage of doctors have average daily working time at least 10.5 hours? (Give the answer correct to 3 significant figures.) (c) Albert thinks that the minimum possible average daily working time of doctors is 1.5 hours. Do you agree? Explain your answer.

Example 6T Solution: 6 More about Statistical Diagrams and Graphs Working time (hours) Class mark (hours) Frequency 3 – 4 3.5 10 5 – 6 5.5 13 7 – 8 7.5 20 9 – 10 9.5 22 11 – 12 11.5 35 13 – 14 13.5 17 (b) Required percentage (cor. to 3 sig. fig.) (c) No. The minimum possible average daily working time is 2.5 hours.

6.2 Histograms, Frequency Polygons and Frequency Curves C. Frequency Curves

Example 7T Solution: 6 More about Statistical Diagrams and Graphs The following graph shows the test marks of Chinese and English for S.2A. In which subject do most students score higher than 80? In which subject do most students score lower than 60? (c) In which subject do the students perform better? Solution: (a) English (b) Chinese (c) The frequency curve for English lies to the right of the curve for Chinese. This means that students get higher marks in English than Chinese. Thus, students perform better in English.

6.3 Cumulative Frequency Polygons and Cumulative Frequency Curves A. Cumulative Frequency Tables

6.3 Cumulative Frequency Polygons and Cumulative Frequency Curves B. Cumulative Frequency Polygons

6.3 Cumulative Frequency Polygons and Cumulative Frequency Curves B. Cumulative Frequency Polygons

Example 8T Solution: 6 More about Statistical Diagrams and Graphs 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 are less than 80 marks. (Give the answer correct to 3 significant figures.) Solution: (a) There are 30 students. (b) Required percentage (cor. to 3 sig. fig.)

Example 8T Solution: 6 More about Statistical Diagrams and Graphs The cumulative frequency polygon shows the marks of a Mathematics test for S.2A. (c) If Betty obtained the 8th lowest marks, find the marks of Betty from the graph. (d) A student will obtain grade C or above if the student obtains x marks or above. It is given that half of the students obtained grade C or above. Find the minimum value of x. Solution: (c) The marks of Betty is 50. (d) From the graph, 15 students obtain 60 marks or above. Therefore, the minimum value of x is 60.

6.3 Cumulative Frequency Polygons and Cumulative Frequency Curves C. Cumulative Frequency Curves

Example 9T Solution: 6 More about Statistical Diagrams and Graphs The following cumulative frequency curve shows the alcoholic concentration (in %) of some drinks. (a) How many drinks are there? (b) How many drinks contain less than 6% alcohol? (c) The alcoholic concentration of the highest 10% of the drinks are brand A. What is the minimum alcoholic concentration of band A drinks? Solution: (a) There are 40 drinks. (b) Number of drinks (c) The required minimum concentration is 8%.

6.3 Cumulative Frequency Polygons and Cumulative Frequency Curves D. Percentiles, Quartiles and Median The pth percentile of a set of data is the number such that p percent of the data is less than that number.

Example 10T Solution: 6 More about Statistical Diagrams and Graphs The following cumulative frequency polygon shows the lengths (in cm) of some springs. (a) What are the lower quartile and upper quartile? (b) Find the 50th percentile. 24 Solution: 8 (a) From the graph, the total frequency is 32. The corresponding cumulative frequency of the lower quartile From the graph, the lower quartile The corresponding cumulative frequency of the upper quartile From the graph, the upper quartile

Example 10T Solution: 6 More about Statistical Diagrams and Graphs The following cumulative frequency polygon shows the lengths (in cm) of some springs. (a) What are the lower quartile and upper quartile? (b) Find the 50th percentile. 16 Solution: (b) The corresponding cumulative frequency of the 50th percentile From the graph, 50th percentile

6.4 Abuses of Statistics

Example 11T Solution: 6 More about Statistical Diagrams and Graphs The bar chart shows the passing rates of S.2 classes 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.2C and S.2D? (c) Does the diagram mislead readers? Explain your answer. Solution: (a) The required ratio (b) From the bar chart, the required ratio (c) Since the ratio of the heights of the bars is different from the ratio of the actual passing rates, the diagram misleads readers. It gives the wrong impression that the passing rate of S.2D is significantly higher than that of S.2C.

Example 12T Solution: 6 More about Statistical Diagrams and Graphs 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) Ratio of the areas of the figures  4 : 1 Ratio of the numbers of readers who read fiction to non-fiction Therefore, the difference in the widths of the figures exaggerates the ratio of the actual numbers of readers. (b) Redraw the diagrams with the same width.

Example 13T Solution: 6 More about Statistical Diagrams and Graphs 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. Solution: (a) Since we don’t know the total monthly expenditures of Mark and Ada, we cannot compare the amounts they spent on different areas.

Example 13T Solution: 6 More about Statistical Diagrams and Graphs The pie charts show the monthly expenditures of Mark and Ada. (b) Suppose that Mark and Ada spent the same amount on food. Find the ratio of their total expenditures. Solution: (b) Let $x and $y be the monthly expenditures of Mark and Ada respectively.  The required ratio