histogram, frequency polygon and frequency curve · a histogram is a graphical representation of...

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Histogram, Frequency Polygon

and Frequency Curve

Frequency Distribution Table for Continuous Data

Histogram

Frequency Polygon and Frequency Curve


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How do we organize the continuous data?

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We can group continuous data into classes

and construct a frequency distribution table

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(1) Class interval The range of each class.

The end values of each class interval,

including lower class limit and upper class

limit.(3) Class mark The mid-value of each class interval.

(4) Lower class boundary The lowest value 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 The difference between the upper and the

lower class boundaries of a class interval.

The highest value 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

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Lower

class

limit

Upper

class

limit


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Consider the class intervals

30 s – 39 s, 40 s – 49 s, 50 s – 59 s, 60 s – 69 s, …

30 39 40 49 50 59 60 69 Time (s)

Lower class

boundary: 39.5

Upper class

boundary: 49.5

Class mark:

44.5

Lower

class

limit

Consider the second class interval:

Upper

class

limit

Consider the third class interval:

Lower class

boundary: 49.5

Upper class

boundary: 59.5

Class mark:

54.5

Note:

The upper boundary of a class interval

is the lower class boundary of the next

class interval.

Histogram

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What is a histogram?

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A histogram is a graphical representation of

continuous data. If all the class widths are equal,

then the frequency of each class interval is

represented by the height of the corresponding bar.

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 axeson a graph paper, then set their scales.

Step 3 Draw the bars of the corresponding classes withheights equal to the frequencies.

Step 4 Give a title 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 on the horizontal axis.

1111


What are frequency polygon and frequency curve?

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A frequency polygon is also a graphical

representation of continuous data. It is

constructed as joining the points with line

segments.

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.

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. Join the adjacent points with line segments.

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 the frequency polygon,

we obtain a frequency curve.

1.345 1.845

Note:

A frequency curve may not pass

through all vertices of its

corresponding frequency polygon.

1616


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

1818


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

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