Young Won Lim11/13/10

● Absolute Frequency● Relative Frequency● Accumulative Frequency● Average● Normalization

Histogram (1A)

Young Won Lim11/13/10

Copyright (c) 2008 Young W. Lim.

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Histogram 3 Young Won Lim11/13/10

Absolute Frequency

math score

frequ

en

cy

4

8

12

16

Average Score?

math score # of students

40 ~ 50 1

50 ~ 60 4

60 ~ 70 8

70 ~ 80 5

80 ~ 90 2

Total 20

40 50 60 70 80 90

45×1 55×4 65×8 75×5 85×21 4 8 5 2

= 66.5

Histogram 4 Young Won Lim11/13/10

Absolute Frequency & Average - Rectangles

1 student

4 students

8 students

5 students

2 students

math score

freq

uency

4

8

12

16

40 50 60 70 80 9010 20 30 100

cum

ula

tive

freq

uen

cy

4

8

12

16

20

math score40 50 60 70 80 9010 20 30 100

75 8545 55 65

45 x 1

55 x 4

65 x 8

75 x 5

85 x 2

math score

frequ

en

cy

4

8

12

16

40 50 60 70 80 9010 20 30 100

75 8545 55 65

Histogram 5 Young Won Lim11/13/10

Absolute Frequency & Average

Absolute Frequency

1 student

4 students

8 students

5 students

2 students

Avg = 66.5

math score

freq

uency

4

8

12

16

40 50 60 70 80 9010 20 30 100

cum

ula

tive

freq

uen

cy

4

8

12

16

20

math score40 50 60 70 80 9010 20 30 100

cum

ula

tive

fr

eq

uen

cy

4

8

12

16

20

math score40 50 60 70 80 9010 20 30 100

math score

freq

uen

cy

4

8

12

16

40 50 60 70 80 90

75 8545 55 65

45 x 1

55 x 4

65 x 8

75 x 5

85 x 2

66.5 x 20

Histogram 6 Young Won Lim11/13/10

Comparison of Histograms – Absolute Frequency

math score # of students

40 ~ 50 1

50 ~ 60 4

60 ~ 70 8

70 ~ 80 5

80 ~ 90 2

Total 20

math score # of students

40 ~ 50 1

50 ~ 60 2

60 ~ 70 8

70 ~ 80 9

80 ~ 90 5

Total 25

Group A

Group B

math score

ab

solu

te f

req

uen

cy

4

8

12

16

40 50 60 70 80 90

math score

ab

solu

te f

req

uen

cy

4

8

12

16

40 50 60 70 80 90

Group A

Group B

Histogram 7 Young Won Lim11/13/10

Normalization

Group A (20 students)

Group B (25 students)

1 4 8 5 2

1 2 8 9 5

A common scale makes comparison convenient

f i =n i

N=

n i

∑knk

relative count

total counts

(absolute) count

Normalization

f i

n i

N =∑ j n j

1.0

1.0

Total 20 student counts

Total 25 student counts

Histogram 8 Young Won Lim11/13/10

Relative Frequency

f i =ni

N=

ni

∑ini

Normalization

f ini

N =∑i ni = 20

math score

40 ~ 50 1 1/20 = 0.05

50 ~ 60 4 4/20 = 0.20

60 ~ 70 8 8/20 = 0.40

70 ~ 80 5 5/20 = 0.25

80 ~ 90 2 2/20 = 0.10

Total 20 20/20 = 1.00

absolutefrequency

relativefrequency math score

40 ~ 50 1 1/25 = 0.04

50 ~ 60 2 2/25 = 0.08

60 ~ 70 8 8/25 = 0.32

70 ~ 80 9 9/25 = 0.36

80 ~ 90 5 5/25 = 0.20

Total 25 25/25 = 1.00

absolutefrequency

relativefrequency

Group A Group B f ini

N =∑i ni = 25

.05 .20 .40 .25 .10 .04 .08 .32 .36 .20

∑i f i = 1.00 ∑i f i = 1.00

Histogram 9 Young Won Lim11/13/10

Comparison of Histograms – Relative Frequency

math score

.2

.4

.6

.8

40 50 60 70 80 90

1.0

math score

.2

.4

.6

.8

40 50 60 70 80 90

1.0

math score

.2

.4

.6

.8

40 50 60 70 80 90

.05

.20

1.0

math score

.2

.4

.6

.8

40 50 60 70 80 90

1.0f ini

N =∑i ni = 20

math score

40 ~ 50 1 1/20 = 0.05

50 ~ 60 4 4/20 = 0.20

60 ~ 70 8 8/20 = 0.40

70 ~ 80 5 5/20 = 0.25

80 ~ 90 2 2/20 = 0.10

Total 20 20/20 = 1.00

absolutefrequency

relativefrequency

Group A

math score

40 ~ 50 1 1/25 = 0.04

50 ~ 60 2 2/25 = 0.08

60 ~ 70 8 8/25 = 0.32

70 ~ 80 9 9/25 = 0.36

80 ~ 90 5 5/25 = 0.20

Total 25 25/25 = 1.00

absolutefrequency

relativefrequency

Group B f ini

N =∑i ni = 25

rela

tive f

req

uen

cy

rela

tive f

req

uen

cy

Group A

Group B

Histogram 10 Young Won Lim11/13/10

Relative Frequency & Average

Relative Frequency

1/20

4/20

8/20

5/20

2/20

Avg = 66.5

math score

rela

tive f

req

uency

.2

.4

.6

.8

40 50 60 70 80 9010 20 30 100

cum

ula

tive

rela

tive f

req

uency

.2

.4

.6

.8

1.0

math score40 50 60 70 80 9010 20 30 100

cum

ula

tive

re

lati

ve f

req

uency

.2

.4

.6

.8

1.0

math score40 50 60 70 80 9010 20 30 100

math score

rela

tive f

req

uency

.2

.4

.6

.8

40 50 60 70 80 90

75 8545 55 65

45 x .05

55 x .20

65 x .40

75 x .25

85 x .10

66.5

1.0

Group A

1.0

Histogram 11 Young Won Lim11/13/10

Average

math score

4

8

12

16

Average Score?

40 50 60 70 80 90

45×1 55×4 65×8 75×5 85×21 4 8 5 2

f ini

N =∑i ni = 20

math score

40 ~ 50 1 1/20 = 0.05

50 ~ 60 4 4/20 = 0.20

60 ~ 70 8 8/20 = 0.40

70 ~ 80 5 5/20 = 0.25

80 ~ 90 2 2/20 = 0.10

Total 20 20/20 = 1.00

absolutefrequency

relativefrequency

Group A

math score

.2

.4

.6

.8

40 50 60 70 80 90

.05

.20

1.0

math score

.2

.4

.6

.8

40 50 60 70 80 90

1.0

=45×1

20

55×420

65×8

20

75×520

85×2

20

= 45×120

55×420

65×820

75×520

85×220

= 66.5

rela

tive f

req

uen

cy

ab

solu

te f

req

uen

cy

Group A Group A

Histogram 12 Young Won Lim11/13/10

Accumulative Frequency

ni

N =∑i ni = 20

math score

40 ~ 50 1 1

50 ~ 60 4 5

60 ~ 70 8 13

70 ~ 80 5 18

80 ~ 90 2 20

Total 20 20

absolutefrequency

accumulativeabsolute frequency

Group A

f i

1.0 =∑i f i = 25

math score

40 ~ 50 0.05 0.05

50 ~ 60 0.20 0.25

60 ~ 70 0.40 0.65

70 ~ 80 0.25 0.90

80 ~ 90 0.10 1.00

Total 1.00 1.00

relativefrequency

accumulativerelative frequency

Group A

math score

4

8

12

16

40 50 60 70 80 901

4

8

5

220

18

13

5

1

math score

.2

.4

.6

.8

40 50 60 70 80 90

.05

.20

.40

.25

.101.0

.90

.65

.25

.05

20

1.0

accu

mu

lat i

ve

rela

tive f

req

uen

cy

accu

mu

lat i

ve

ab

solu

te f

req

uen

cy

Group A

Group A

Histogram 13 Young Won Lim11/13/10

Relative Frequency & Ratio

2 student

5 students

8 students

4 students

1 students

Absolute frequency

.2

.4

.6

.8

.4 .5 .6 .7 .8 .9.1 .2 .3

1.0

Relative frequency

20

.10

.25

.40

.20

.05

f ini

N =∑i ni = 20

math score

40 ~ 50 1 1/20 = 0.05

50 ~ 60 4 4/20 = 0.20

60 ~ 70 8 8/20 = 0.40

70 ~ 80 5 5/20 = 0.25

80 ~ 90 2 2/20 = 0.10

Total 20 20/20 = 1.00

absolutefrequency

relativefrequency

Group A

Histogram 14 Young Won Lim11/13/10

Normalization Equations (1)

f i =n i

N

0.xxx =△△

□□□□□

fraction =part

total

relativefrequency

count

total count=

N =n i

f i

□□□□□ =△△

0.xxx

totalpart

fraction

totalcount

count

relative freq=

n i = N ⋅ f i

□□□□□△△ 0.xxx

totalcount

count relativefrequency

=

=

•

totalpart part= •=

•

Histogram 15 Young Won Lim11/13/10

f i =n i

N

N =n i

f i n i = N ⋅ f i

fi

ni

N1.0

fi

ni

N1.0

fi

ni

N1.0

fi N

ni

= slope

= slope= slope

Normalization Equations (2)

Young Won Lim11/13/10

References

[1] http://en.wikipedia.org/[2] http://planetmath.org/[3] “ ”최상위수학 디딤돌[4] “ ”센 수학[5] “ ” 개념과 유형 비상