histogram(組織圖), frequency polygon(頻數多邊形) and …

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Histogram(組織圖), Frequency Polygon(頻數多邊形) and

Frequency Curve (頻數曲線)

Frequency Distribution Table (頻數分佈表) for Continuous Data

Histogram

Frequency Polygon and Frequency Curve

Frequency Distribution Table for Continuous Data

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How do we

organize(組織) the

continuous data?

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We can group (分組) continuous data into

classes and construct a frequency

distribution table (頻數分佈表) to organize

the data.


Some useful information should be included

in the frequency distribution table.

Let’s discuss some of the terms.


Term Explanation

(2) Class limits組限

(1) Class interval組區間 The range範圍 of each class.

The end values 兩個末端的值 of each class

interval, including lower class limit下組限and upper class limit上組限.

(3) Class mark組中點 The mid-value中間的值 of each class

interval.

(4) Lower class boundary

下組界

The lowest value 最小值of a class interval.

It is the mid-value of the lower class limit

and the upper class limit of the previous class.

(6) Class width組距The difference差 between the upper and the

lower class boundaries of a class interval.

The highest value 最大值 of a class interval.

It is the mid-value of the upper class limit

and the lower class limit of the next class.

(5) Upper class boundary

上組界

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Histogram

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What is a histogram

組織圖?

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A histogram is a graphical representation 圖像方式表示 of continuous data. If all the class widths are

equal 組距相等, then the frequency 頻數of each

class interval is represented by the height高度of the

corresponding bar對應長方形棒條.

Histogram

Here are the steps for drawing a histogram.

Step 1 Construct a suitable frequency distribution table.

Step 2 Properly label the horizontal and the vertical axes

on a graph paper, then set their scales.

Step 3 Draw 繪畫 the bars of the corresponding classes

with heights高 equal to等於 the frequencies頻數.

Step 4 Give a title標題 to the histogram.

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Example 1

Histogram

The heights (in m) of a class of students are listed below.

1.62 1.70 1.73 1.66 1.40

1.47 1.60 1.79 1.71 1.41

1.67 1.56 1.53 1.54 1.73

1.78 1.50 1.62 1.76 1.64

(a) Construct a frequency distribution table for the above data.

Use 1.40 m – 1.49 m as the first class interval, 1.50 m – 1.59 m

as the second class interval and so on.

(b) Draw a histogram to present the frequency distribution.

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Example 1

Histogram

(a)

The heights (in m) of a class of students are listed below.

1.62 1.70 1.73 1.66 1.40

1.47 1.60 1.79 1.71 1.41

1.67 1.56 1.53 1.54 1.73

1.78 1.50 1.62 1.76 1.64

Height (m)Class

boundaries (m)

Class

mark (m)Tally Frequency

1.40 – 1.49

1.50 – 1.59

1.60 – 1.69

1.70 – 1.79

1.395 – 1.495 1.445

1.495 – 1.595 1.545

1.595 – 1.695 1.645

1.695 – 1.795 1.745

3

4

6

7

Total 20

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Example 1

(b)

Histogram

Heights of a class of students

01.445 1.545 1.645 1.745

1

2

3

4

5

6

7

Height (m)

Fre

quen

cy

Label the class marks 組中點on the horizontal

axis.

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What are frequency polygon

頻數多邊形and frequency curve

頻數曲線?

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A frequency polygon 頻數多邊形 is also a

graphical representation 圖像方式表示 of

continuous data. It is constructed as joining the

points連接相鄰的點 with line segments線段.

Here are the steps for drawing a frequency polygon.

Step 1 Construct a suitable frequency distribution table

where the frequencies of the first and the last class

marks are 0 第一個組中點和最後一個組中點的頻數為零.

Step 2 Properly label the horizontal and the vertical axes

on a graph paper, then set their scales.

Step 3 Plot frequencies against class marks標出頻數對組中點的點. Join the adjacent points with line

segments 把相鄰的點用線段連接.

Step 4 Give a title to the frequency polygon.

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Let’s discuss the data in Example 1 again.

For drawing a frequency polygon or a frequency

curve, two additional class marks with frequencies

zero must be added to the original leftmost and

rightmost class marks.

1.30 – 1.39 1.345 0

1.80 – 1.89 1.845 0

Height (m) Class mark (m) Frequency

1.40 – 1.49

1.50 – 1.59

1.60 – 1.69

1.70 – 1.79

1.445 3

4

6

7

Total 20

1.545

1.645

1.745


01.445 1.545 1.645 1.745

1

2

3

4

5

6

7

Height (m)

Fre

quen

cy

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Plot the points (class mark, frequency) of each class

interval on the graph.

Then join the adjacent points with line segments and we

obtain a frequency polygon.

1.345 1.845


01.445 1.545 1.645 1.745

1

2

3

4

5

6

7

Height (m)

Fre

quen

cy

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By smoothing 平滑 the frequency polygon, we

obtain a frequency curve 頻數曲線.

1.345 1.845

Note:

A frequency curve may not 不一定pass through 通過 all vertices 所有頂點 of its corresponding frequency

polygon.

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Example 2

The data below show the weights of gold nuggets (measured in g)

collected by a gold miner on a certain day:

0.53 0.46 0.52 0.56 0.51 0.41 0.52

0.55 0.48 0.50 0.55 0.55 0.57 0.50

0.47 0.55 0.51 0.50 0.42 0.49 0.46

0.43 0.54 0.51 0.49 0.55 0.47 0.41

(a) Construct a frequency distribution table for the above data.

Use 0.41 g – 0.45 g as the first class interval, 0.46 g – 0.50 g

as the second class interval and so on.

(b) Draw a frequency polygon to present the frequency

distribution.

(c) Draw a frequency curve to present the frequency

distribution.

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Example 2

(a)

The data below show the weights of gold nuggets (measured in g)

collected by a gold miner on a certain day:

0.53 0.46 0.52 0.56 0.51 0.41 0.52

0.55 0.48 0.50 0.55 0.55 0.57 0.50

0.47 0.55 0.51 0.50 0.42 0.49 0.46

0.43 0.54 0.51 0.49 0.55 0.47 0.41

Weight (g)Class

boundaries (g)

Class

mark (g)Tally Frequency

0.41 – 0.45

0.46 – 0.50

0.51 – 0.55

0.56 – 0.60

0.405 – 0.455 0.43

0.455 – 0.505 0.48

0.505 – 0.555 0.53

0.555 – 0.605 0.58

4

10

12

2

Total 28

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Example 2

(b)

(c)Weights of gold nuggets

00.38

2

4

6

8

10

12

14

Weight (g)

Fre

quen

cy

0.43 0.48 0.53 0.58 0.63

histogram(組織圖), frequency polygon(頻數多邊形) and …

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