Basic Statistic Measure of Dispersion

7/31/2019 Basic Statistic Measure of Dispersion

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Part 3:

Measure of Dispersion

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Why to analyze measure ofdispersion

To evaluate central tendency of a data set.

To compare distribution of two or more data set.


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Measures of Dispersion

(Range)

(Mean deviation)

(Variance)

(Standard deviation)


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Non-grouped datameasure of dispersion

(Range): The difference between maximum andminimum values in a data serie.

R = Maximum value

Minimum value


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Range

Series 1 Series 2

2 5

3 5

6 57 6

8 7

10 8X = 6 X = 6

R = 10 2 = 8 R = 8 5 = 3

____


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Grouped data setRange

Hourly wage

YTL

Frequency

5-10 10

10-15 21

15-20 9

20-25 5

R = 25 5 = 20 YTL


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(Mean Deviation or Mean Absolute Deviation)

Absolute deviations of all values in a population

from the populations aritmetic mean.

X- X

M.D. = N

___________

__

X X = absolute deviation__


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Mean deviation

(Non-grouped data)

Calculate absolute deviation of the following value serie:

15, 16, 18, 21, 25

16TOTAL

625-1925

221-1921

118-1918

316-1916

415-1915X - XX - XValues

____

16

M.D. = = 3.2

5

____


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Mean Deviation

(Grouped data set)

486.2169551Total

10921.855-33.227555550-60

106.211.845-33.240545940-50

28.81.835-33.2560351630-40

114.88.225-33.2350251420-30

127.418.215-33.210515710-20

f X XX - XX - X

fXGroup Mid-

point ( X)

Frequency

( f)

Group _ __

f X - X 486.2M.D. = = = 9.53

N 51

___________ _____

_

fX 1695X= = = 33.2

N 51

___________


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(Variance and Standard Deviation)

Variance , is the square aritmetic mean of all

deviations of values from the mean)

( X X )

=

N

__


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Standard Deviation

is the square root of variance of a data set.

( X X )

=

N

__


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Characteristics of Standard Deviationand Variance

( cX ) = c ( X )

( cX ) = c ( X )

( X a ) = ( X )

( X + a ) = ( X )


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Standard Deviation and Variance(Non-grouped data)

Calculate standard deviation and variance of the following

values: 22, 25, 28, 30 ve 35

( X X ) = 98( X X ) = 0 X = 140

49735

4230

0028

9-325

36-622

( X X )( X X )X__

__

98

= = 19.65

= 19.6 = 4.43

140

X = = 285

_


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Standard Deviation and Variance(Grouped data set)

14088001180032Total

65340010890033042007006600-800

1183001690013035005007400-600

539004900-70330030011200-400

58320072900-27080010080-200

f (X X)(X X)(X X)

fXGroup Mid-

point ( X)

Frequen.

( f)

groups _ __

1408800

= = 44025

32

_______11800

X= = 370

32

______

= 44025 = 209.82


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Variance

Its difficult to comment on asingle data set. Likely

average mean deviaiton or range, variance is

used to compare variation in two data sets.


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Relation between standard deviation and aritmeticmean in symmetric frequency distributions


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Relative Dispersion

Coeffient of variation : is expressed in %. Standard

deviation divided by aritmetic mean.

to benchmark two data sets which have different units (cm, Rp.)

to compare data which have same units same but means are

very different

COV (%) = * 100X

_

X = 500000, = 50000

X = 12000, = 2000_

_50000

COV = * 100 = 10%

500000

2000

COV = * 100 = 16.7%

12000


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Example (Relative Dispersion)

For a product A

T1 machine T2 machine

Length mean (X1) = 67 mm Length X2 = 64 mm

Standard deviation (1) = 2.5 mm 2 = 2.4 mm

X1 > X2 1 > 2 (as seen). It can be said that theres

much variation in first machine, however this comment is

wrong.

V1 = (2.5 / 67) * 100 = % 3.73 In fact, much variation in machine

V2 = (2.4 / 64) * 100 = % 3.75 T2

_

_

COV (%) = * 100X_

_

_


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Skewness

Expresses the skewness grade of frequency distributions.

Skewness is zero at symmetric frequency distributions.

Skewness usual vary between -3 and +3 .

3 (X median)Skewness =

_


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Skewness

Negative

Skewness

Positive

Skewness

Symmetric

Skewness = 0

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Basic Statistic Measure of Dispersion