1
Let’
s h
ave
a M
ath
Ch
at
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?
3
We can group (分組) continuous data into
classes and construct a frequency
distribution table (頻數分佈表) to organize
the data.
Frequency Distribution Table for Continuous Data
Some useful information should be included
in the frequency distribution table.
Let’s discuss some of the terms.
Frequency Distribution Table for Continuous Data
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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Frequency Polygon and Frequency Curve
What are frequency polygon
頻數多邊形and frequency curve
頻數曲線?
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Frequency Polygon and Frequency Curve
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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Frequency Polygon and Frequency Curve
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
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
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Frequency Polygon and Frequency Curve
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
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
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Frequency Polygon and Frequency Curve
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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Frequency Polygon and Frequency Curve
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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Frequency Polygon and Frequency Curve
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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Frequency Polygon and Frequency Curve
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