construction and interpretation of simple diagrams and graphs (ii) 6 6.1organization of continuous...

Construction and Interpretation of Simple Diagrams and Graphs (II)

6

6.1 Organization of Continuous Data6.2 Histogram, Frequency Polygon and

Frequency Curve6.3 Cumulative Frequency Polygon and

Cumulative Frequency Curve

Chapter Summary

Mathematics in Workplaces

6.4 Abuses of Statistics

P. 2

NutritionistGrowth 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

P. 3

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.

Table 6.2Lower class limit

Upper class limit

12//// //// //20 – 24

10//// ////15 – 19

5////10 – 14

14 //// //// ////5 – 9

FrequencyTallyTime spent (min)

12 min

min 2

1410

Class mark

2

limit classUpper limit classLower

P. 4

Class width Upper class boundary – Lower class boundary


(14.5 – 9.5) min 5 min

12//// //// //20 – 24

10//// ////15 – 19

5////10 – 14

14 //// //// ////5 – 9

FrequencyTallyTime spent (min)

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.

2436172018155Frequency

75–7970–7465–6960–6455–5950–5445–49Average Speed (km/h)


Solution:

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 1.745

m.


Example 6.2T

1.81 1.55 1.69 1.59 1.74 1.79 1.60 1.751.73 1.56 1.66 1.58 1.78 1.59 1.67 1.681.71 1.72 1.80 1.65 1.73 1.70 1.61 1.771.68 1.69 1.57 1.66 1.75 1.58 1.74 1.761.75 1.77 1.59 1.78 1.76 1.64 1.62 1.69

Solution:

(b) Class width (1.595 – 1.545) m

(a) Click here: Frequency distribution table

(c) Required percentage

30% 0.05 m

%10040

210

P. 7

6.2 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 – 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

A. Histogram

P. 8

Height (cm) Class mark (cm)

Class boundaries (cm) Frequency

20 – 29 24.5 19.5 – 29.5 15

Table 6.16

Fig. 6.5

A. Histogram


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

P. 9

The following shows the mobile phone monthly usage (in minutes) of some businessmen.

51599.5 – 1899.51600 – 1899

121299.5 – 1599.51300 – 1599

35999.5 – 1299.51000 – 1299

28699.5 – 999.5700 – 999

23399.5 – 699.5400 – 699

1099.5 – 399.5100 – 399

FrequencyClass boundaries (in minutes)Usage (minutes)

Construct a histogram for the above data.

A. Histogram


Example 6.3T

P. 10

A. Histogram


Example 6.3T

Solution:

P. 11

A histogram looks similar to a bar chart. However, there are some differences between them.

Table 6.19

A. Histogram


NoYesWith gaps between bars

For continuous dataFor discrete dataNature of data

Histogram Bar chart

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. 6.30


B. Frequency Polygon

P. 13

The following table shows the areas (in cm2) of 50 tiles in different shapes.

15130.5121 – 140

8110.5101 – 120

690.581 – 100

370.561 – 80

650.541 – 60

1230.521 – 40

FrequencyClass mark (cm2)Area (cm2)

Construct a frequency polygon for the above data.



Example 6.4T

P. 14



Example 6.4T

Solution:

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.



Example 6.5T

Solution:(b) The class interval ‘10 hours – 11

hours’.(c) Number of doctors work more than 10 hours 35 17(d) Number of doctors 10 13 20 22 35

17 117

52

Frequency distribution table

1013

2022

35

17

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. 6.11

The curve does not necessarily pass through all the vertices of the frequency polygon.


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


(a) English

Solution:

Example 6.6T

P. 18

6.3 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.

MarksClass

boundariesNumber 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 5

90 – 99 89.5 – 99.5 3

Marksless than

Number of students

39.5 0

99.5 37 + 3 40

49.5 5

59.5 5 + 6 11

69.5 11 + 8 19

79.5 19 + 13 32

89.5 32 + 5 37

Cumulative frequency tableFrequency distribution table

Table 6.29

P. 19

Fig. 6.23

2. the cumulative frequency must start from zero,

3. the last point on the cumulative frequency polygon refers to

the total number of data.



1. the trend of the graph never goes down since the cumulative frequency never decreases,

From the cumulative frequency polygon, we can see that

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.



Solution:

(a) There are 30 students.

(b) Required percentage %10030

1525

%3

133

(c) Required percentage

%10030

8

%3

226

Example 6.7T

P. 21

B. Cumulative Frequency Curve

Fig. 6.26


Like a frequency curve, we can smooth a cumulative frequency polygon to become a curve called a cumulative frequency curve.

P. 22


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.


1. the trend of the graph never goes down since the cumulative frequency never decreases,

Fig. 6.26

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?



Example 6.8T

Solution:(a) There are 40 drinks.

(b) 16 drinks contain less than 6% alcohol.

(c) Number of drinks 40 28 12

P. 24

We can obtain the corresponding data from a particular number of observations by using percentiles and quartiles.

Fig. 6.30


C. Percentiles, Quartiles and Median

P. 25

C. Percentiles, Quartiles and Median


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

8

24

16

P. 26


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.

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.


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.

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.


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.

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.


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,

90

360:

180

360required ratio

2:1

P. 30

Chapter Summary


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. 2

limit classUpper limit classLower mark Class

P. 31


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

P. 32


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.

P. 33

Chapter Summary


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.

P. 34


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.

construction and interpretation of simple diagrams and graphs (ii) 6 6.1organization of continuous...

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